The Tate-curve index conjecture for SQFTs with cylindrical ends

Let $\mathcal F$ be an $\mathcal N=(0,1)$ SQFT with cylindrical ends and boundary $\partial\mathcal F$, satisfying $Z_{RR}(\partial\mathcal F)=0$, and suppose $\partial\mathcal F$ is superconformal. Let $\mathcal H(\mathcal F)$ be the $S^1$-equivariant SQM obtained by compactifying $\mathcal F$ on the Ramond circle called the Tate curve. Tate-curve index conjecture. The holomorphic part of $Z_{RR}(\mathcal F)$ exists, and its $q$-expansion is the index of $\mathcal H(\mathcal F)$, lying in $\mathbb Z((q))$ up to an Atiyah–Patodi–Singer correction. This correction is a half-integer related to a mod-2 index of $\partial\mathcal F$. The conjecture connects conditionally convergent SQFT partition functions to mock modular forms. The paper attributes the proposal to the analysis of GJFmock and does not claim a general rigorous proof.