Complex-conjugation relation between twisted modular data

Establish that the modular data obtained from the A- and B-twisted sectors of the DGG theories associated with orientation-reversal-invariant punctured lens spaces $L_{n,k}$ are related by complex conjugation.

Background

The paper argues that orientation reversal of the punctured lens space corresponds to parity conjugation of the associated three-dimensional gauge theory and hence to the mirror action on the axial symmetry. It further suggests that the modular data produced by the A- and B-twisted sectors should exhibit a corresponding complex-conjugation relationship, but this claim is stated as a conjecture rather than proved.