Essential virtual pairs: $S: BQ^4_{45} \quad U = ((1 2) , (1 2)(3 4) , (3 4) , (3 4)) \quad D = ((1 2) , (1 2)(3 4) , (3 4) , (3 4))$\hfil\break $T: BQ^4_{1} \quad U = (\iota , \iota , (3 4) , (3 4)) \quad D = (\iota , \iota , (3 4) , (3 4))$\hfil\break $S: BQ^4_{46} \quad U = ((1 2) , (1 2)(3 4) , (3 4) , (3 4)) \quad D = ((1 2)(3 4) , (1 2) , (3 4) , (3 4))$\hfil\break $T: BQ^4_{1} \quad U = (\iota , \iota , (3 4) , (3 4)) \quad D = (\iota , \iota , (3 4) , (3 4))$\hfil\break $S: BQ^4_{3} \quad U = (\iota , \iota , (2 4 3) , (2 3 4)) \quad D = (\iota , (3 4) , (3 4) , (3 4))$\hfil\break $T: BQ^4_{30} \quad U = ((3 4) , (3 4) , (3 4) , (3 4)) \quad D = ((3 4) , (3 4) , (3 4) , (3 4))$\hfil\break $S: BQ^4_{42} \quad U = ((1 2) , (1 2) , (3 4) , (3 4)) \quad D = ((1 2) , (1 2) , (3 4) , (3 4))$\hfil\break $T: BQ^4_{9} \quad U = (\iota , (3 4) , (3 4) , (3 4)) \quad D = (\iota , (3 4) , (3 4) , (3 4))$\hfil\break $S: BQ^4_{43} \quad U = ((1 2) , (1 2) , (3 4) , (3 4)) \quad D = ((1 2)(3 4) , (1 2)(3 4) , (3 4) , (3 4))$\hfil\break $T: BQ^4_{9} \quad U = (\iota , (3 4) , (3 4) , (3 4)) \quad D = (\iota , (3 4) , (3 4) , (3 4))$\hfil\break $S: BQ^4_{14} \quad U = (\iota , (3 4) , (1 4 3) , (1 3 4)) \quad D = ((3 4) , (3 4) , (3 4) , (3 4))$\hfil\break $T: Q^4_{1} \quad U = \iota \quad D = \iota$\hfil\break $S: BQ^4_{15} \quad U = (\iota , (2 3) , (3 4) , (2 4)) \quad D = (\iota , (2 3 4) , (2 3 4) , (2 3 4))$\hfil\break $T: BQ^4_{16} \quad U = (\iota , (2 3 4) , (2 3 4) , (2 3 4)) \quad D = (\iota , (2 4 3) , (2 4 3) , (2 4 3))$\hfil\break $S: BQ^4_{15} \quad U = (\iota , (2 3) , (3 4) , (2 4)) \quad D = (\iota , (2 3 4) , (2 3 4) , (2 3 4))$\hfil\break $T: BQ^4_{39} \quad U = ((2 3 4) , (2 3 4) , (2 3 4) , (2 3 4)) \quad D = ((2 4 3) , (2 4 3) , (2 4 3) , (2 4 3))$\hfil\break $S: BQ^4_{15} \quad U = (\iota , (2 3) , (3 4) , (2 4)) \quad D = (\iota , (2 3 4) , (2 3 4) , (2 3 4))$\hfil\break $T: BQ^4_{40} \quad U = ((2 3 4) , (2 4 3) , (2 4 3) , (2 4 3)) \quad D = ((2 4 3) , (2 3 4) , (2 3 4) , (2 3 4))$\hfil\break $S: BQ^4_{15} \quad U = (\iota , (2 3) , (3 4) , (2 4)) \quad D = (\iota , (2 3 4) , (2 3 4) , (2 3 4))$\hfil\break $T: Q^4_{1} \quad U = \iota \quad D = \iota$\hfil\break $S: BQ^4_{37} \quad U = ((2 3 4) , (2 3) , (3 4) , (2 4)) \quad D = ((2 4 3) , (2 3 4) , (2 3 4) , (2 3 4))$\hfil\break $T: BQ^4_{16} \quad U = (\iota , (2 3 4) , (2 3 4) , (2 3 4)) \quad D = (\iota , (2 4 3) , (2 4 3) , (2 4 3))$\hfil\break $S: BQ^4_{38} \quad U = ((2 3 4) , (2 3 4) , (2 3 4) , (2 3 4)) \quad D = ((2 4 3) , (2 3) , (3 4) , (2 4))$\hfil\break $T: BQ^4_{16} \quad U = (\iota , (2 3 4) , (2 3 4) , (2 3 4)) \quad D = (\iota , (2 4 3) , (2 4 3) , (2 4 3))$\hfil\break $S: BQ^4_{20} \quad U = (\iota , \iota , (3 4) , (1 2)(3 4)) \quad D = (\iota , \iota , (3 4) , (1 2)(3 4))$\hfil\break $T: BQ^4_{25} \quad U = (\iota , \iota , (1 2) , (1 2)) \quad D = (\iota , \iota , (1 2) , (1 2))$\hfil\break $S: BQ^4_{20} \quad U = (\iota , \iota , (3 4) , (1 2)(3 4)) \quad D = (\iota , \iota , (3 4) , (1 2)(3 4))$\hfil\break $T: Q^4_{1} \quad U = \iota \quad D = \iota$\hfil\break $S: BQ^4_{21} \quad U = (\iota , \iota , (1 2)(3 4) , (3 4)) \quad D = (\iota , \iota , (3 4) , (1 2)(3 4))$\hfil\break $T: BQ^4_{25} \quad U = (\iota , \iota , (1 2) , (1 2)) \quad D = (\iota , \iota , (1 2) , (1 2))$\hfil\break $S: BQ^4_{21} \quad U = (\iota , \iota , (1 2)(3 4) , (3 4)) \quad D = (\iota , \iota , (3 4) , (1 2)(3 4))$\hfil\break $T: Q^4_{1} \quad U = \iota \quad D = \iota$\hfil\break $S: BQ^4_{52} \quad U = ((1 2 3 4) , (1 4 3 2) , (1 3) , (2 4)) \quad D = ((1 2 4 3) , (1 3 4 2) , (2 3) , (1 4))$\hfil\break $T: BQ^4_{26} \quad U = (\iota , \iota , (1 2)(3 4) , (1 2)(3 4)) \quad D = (\iota , \iota , (1 2)(3 4) , (1 2)(3 4))$\hfil\break $S: BQ^4_{53} \quad U = ((1 2 3 4) , (1 4 3 2) , (1 2 3 4) , (1 4 3 2)) \quad D = ((1 2 3 4) , (1 4 3 2) , (1 2 3 4) , (1 4 3 2))$\hfil\break $T: BQ^4_{26} \quad U = (\iota , \iota , (1 2)(3 4) , (1 2)(3 4)) \quad D = (\iota , \iota , (1 2)(3 4) , (1 2)(3 4))$\hfil\break $S: BQ^4_{29} \quad U = (\iota , (3 4) , \iota , \iota) \quad D = ((3 4) , (3 4) , (1 4) , (1 3))$\hfil\break $T: Q^4_{1} \quad U = \iota \quad D = \iota$\hfil\break $S: BQ^4_{45} \quad U = ((1 2) , (1 2)(3 4) , (3 4) , (3 4)) \quad D = ((1 2) , (1 2)(3 4) , (3 4) , (3 4))$\hfil\break $T: BQ^4_{30} \quad U = ((3 4) , (3 4) , (3 4) , (3 4)) \quad D = ((3 4) , (3 4) , (3 4) , (3 4))$\hfil\break $S: BQ^4_{46} \quad U = ((1 2) , (1 2)(3 4) , (3 4) , (3 4)) \quad D = ((1 2)(3 4) , (1 2) , (3 4) , (3 4))$\hfil\break $T: BQ^4_{30} \quad U = ((3 4) , (3 4) , (3 4) , (3 4)) \quad D = ((3 4) , (3 4) , (3 4) , (3 4))$\hfil\break $S: BQ^4_{34} \quad U = (\iota , \iota , \iota , (1 3 2)) \quad D = ((2 3) , (1 3) , (1 2) , (1 2 3))$\hfil\break $T: Q^4_{1} \quad U = \iota \quad D = \iota$\hfil\break $S: BQ^4_{35} \quad U = ((3 4) , (1 4 2 3) , (1 2) , (1 3 2 4)) \quad D = ((2 3) , (1 3 4 2) , (1 4) , (1 2 4 3))$\hfil\break $T: Q^4_{1} \quad U = \iota \quad D = \iota$\hfil\break $S: BQ^4_{37} \quad U = ((2 3 4) , (2 3) , (3 4) , (2 4)) \quad D = ((2 4 3) , (2 3 4) , (2 3 4) , (2 3 4))$\hfil\break $T: BQ^4_{39} \quad U = ((2 3 4) , (2 3 4) , (2 3 4) , (2 3 4)) \quad D = ((2 4 3) , (2 4 3) , (2 4 3) , (2 4 3))$\hfil\break $S: BQ^4_{37} \quad U = ((2 3 4) , (2 3) , (3 4) , (2 4)) \quad D = ((2 4 3) , (2 3 4) , (2 3 4) , (2 3 4))$\hfil\break $T: Q^4_{1} \quad U = \iota \quad D = \iota$\hfil\break $S: BQ^4_{38} \quad U = ((2 3 4) , (2 3 4) , (2 3 4) , (2 3 4)) \quad D = ((2 4 3) , (2 3) , (3 4) , (2 4))$\hfil\break $T: BQ^4_{40} \quad U = ((2 3 4) , (2 4 3) , (2 4 3) , (2 4 3)) \quad D = ((2 4 3) , (2 3 4) , (2 3 4) , (2 3 4))$\hfil\break $S: BQ^4_{38} \quad U = ((2 3 4) , (2 3 4) , (2 3 4) , (2 3 4)) \quad D = ((2 4 3) , (2 3) , (3 4) , (2 4))$\hfil\break $T: Q^4_{1} \quad U = \iota \quad D = \iota$\hfil\break $S: Q^4_{3} \quad U = ((2 4 3) , (1 3 4) , (1 4 2) , (1 2 3)) \quad D = \iota$\hfil\break $T: BQ^4_{39} \quad U = ((2 3 4) , (2 3 4) , (2 3 4) , (2 3 4)) \quad D = ((2 4 3) , (2 4 3) , (2 4 3) , (2 4 3))$\hfil\break $S: BQ^4_{41} \quad U = ((2 3 4) , (1 3 2) , (1 4 3) , (1 2 4)) \quad D = ((2 3 4) , (2 3 4) , (2 3 4) , (2 3 4))$\hfil\break $T: Q^4_{1} \quad U = \iota \quad D = \iota$\hfil\break $S: BQ^4_{45} \quad U = ((1 2) , (1 2)(3 4) , (3 4) , (3 4)) \quad D = ((1 2) , (1 2)(3 4) , (3 4) , (3 4))$\hfil\break $T: Q^4_{1} \quad U = \iota \quad D = \iota$\hfil\break $S: BQ^4_{46} \quad U = ((1 2) , (1 2)(3 4) , (3 4) , (3 4)) \quad D = ((1 2)(3 4) , (1 2) , (3 4) , (3 4))$\hfil\break $T: Q^4_{1} \quad U = \iota \quad D = \iota$\hfil\break $S: BQ^4_{52} \quad U = ((1 2 3 4) , (1 4 3 2) , (1 3) , (2 4)) \quad D = ((1 2 4 3) , (1 3 4 2) , (2 3) , (1 4))$\hfil\break $T: BQ^4_{50} \quad U = ((1 2)(3 4) , (1 2)(3 4) , (1 2)(3 4) , (1 2)(3 4)) \quad D = ((1 2)(3 4) , (1 2)(3 4) , (1 2)(3 4) , (1 2)(3 4))$\hfil\break $S: Q^4_{3} \quad U = ((2 4 3) , (1 3 4) , (1 4 2) , (1 2 3)) \quad D = \iota$\hfil\break $T: BQ^4_{50} \quad U = ((1 2)(3 4) , (1 2)(3 4) , (1 2)(3 4) , (1 2)(3 4)) \quad D = ((1 2)(3 4) , (1 2)(3 4) , (1 2)(3 4) , (1 2)(3 4))$\hfil\break $S: BQ^4_{52} \quad U = ((1 2 3 4) , (1 4 3 2) , (1 3) , (2 4)) \quad D = ((1 2 4 3) , (1 3 4 2) , (2 3) , (1 4))$\hfil\break $T: Q^4_{1} \quad U = \iota \quad D = \iota$\hfil\break $S: BQ^4_{53} \quad U = ((1 2 3 4) , (1 4 3 2) , (1 2 3 4) , (1 4 3 2)) \quad D = ((1 2 3 4) , (1 4 3 2) , (1 2 3 4) , (1 4 3 2))$\hfil\break $T: Q^4_{1} \quad U = \iota \quad D = \iota$\hfil\break $S: BQ^4_{55} \quad U = ((1 2)(3 4) , (1 3)(2 4) , (1 3)(2 4) , (1 2)(3 4)) \quad D = ((1 3)(2 4) , (1 2)(3 4) , (1 2)(3 4) , (1 3)(2 4))$\hfil\break $T: Q^4_{1} \quad U = \iota \quad D = \iota$\hfil\break $S: BQ^4_{56} \quad U = ((1 2)(3 4) , (1 2)(3 4) , (1 2)(3 4) , (1 2)(3 4)) \quad D = ((1 3 2) , (1 2 4) , (1 4 3) , (2 3 4))$\hfil\break $T: Q^4_{1} \quad U = \iota \quad D = \iota$\hfil\break