Gukov–Putrov–Vafa conjecture on the superconformal-index factorization

Let $M_3$ be a closed oriented $3$-manifold and let $\mathcal I(q)=\operatorname{Tr}_{\mathcal H_{S^2}}(-1)^Fq^{R/2+J_3}$ be the superconformal index of $T[M_3]$. Let $a$ range over $\operatorname{Tor} H_1(M_3,\mathbb Z)/\mathbb Z_2$, let $\mathcal W_a$ be the stabilizer of $a$ under the Weyl-group action, and let $\widehat Z_a(q)$ be the series appearing in the WRT decomposition.

Index-factorization conjecture. The index satisfies

$$\mathcal I(q)=\sum_{a\in\operatorname{Tor} H_1(M_3,\mathbb Z)/\mathbb Z_2}|\mathcal W_a|\widehat Z_a(q)\widehat Z_a(q^{-1})\in\mathbb Z[[q]],$$

where $\widehat Z_a(q^{-1})$ is an appropriate extension of $\widehat Z_a(q)$ to $|q|>1$.

The conjecture expresses the index as a pairing of the homological blocks associated with opposite $q$-domains and predicts integral coefficients. The source explains that orientation reversal can define the extension in general, but gives no resolution of the conjecture.