Self-mirror characterization of DGG rank-0 SCFTs

Prove that the DGG theory $T[L_{n,k}]$ associated with the punctured lens space $L(2n+3,2k)\setminus\{v\}$ is a self-mirror rank-0 SCFT if and only if $2k^2\equiv n+1\pmod{2n+3}$.

Background

Orientation reversal sends a lens space $L(p,q)$ to $L(p,-q)$, while lens-space homeomorphism identifies $L(p,q)$ with $L(p,q^{-1})$. Consequently, orientation-reversal invariance is equivalent to $q^2\equiv-1\pmod p$, which becomes $2k^2\equiv n+1\pmod{2n+3}$ for the punctured lens spaces considered here. The paper conjectures that this topological symmetry induces the three-dimensional $\mathcal N=4$ mirror symmetry of the associated rank-0 SCFT. Equality of superconformal indices is checked only to finite order for several examples.