Vertical asymptotics conjecture for the meromorphic 3D-index

Let $I^{\mathrm{mer}}(0,0)$ be the meromorphic 3D-index of a hyperbolic knot, let $\sigma$ range over the relevant boundary-parabolic $\operatorname{PSL}_2(\mathbbm C)$-representations, and let $\widehat{\Phi}^{(\sigma)}_n$ and $\widehat{\Phi}^{(\sigma)}(x,\hbar)$ be the associated discrete and $x$-deformed asymptotic series. For each $\sigma$, let $C_\sigma$ be the contour appearing in the period contribution. Meromorphic vertical asymptotics conjecture. When $\tau\downarrow0$ vertically,

$$I^\mathrm{mer}(0,0)(e^{2 \pi i \tau}) \sim \sum_{\sigma\,\,\mathrm{complex}} \sum_{n \in \mathbbm Z} \widehat{\Phi}^{(\sigma)}_n(2\pi i \tau) \widehat{\Phi}^{(\bar\sigma)}_n(-2\pi i \tau) + \frac{1}{2\pi i \tau} \sum_{\sigma} \int_{C_\sigma} \widehat{\Phi}^{(\sigma)}(e^{\tau u},2\pi i \tau) \widehat{\Phi}^{(\sigma)}(e^{\tau u},-2\pi i \tau)\,du.$$

The second term consists of period contributions associated with branches of the $A$-polynomial curve, explaining why periods rather than only algebraic coefficients occur in the vertical asymptotics of the meromorphic index. The claim is presented as a conjectural asymptotic formula and remains open.