Twisted-index integrality conjecture for adjoint Reidemeister torsions

Let $M$ be a compact 3-manifold with a torus boundary, and suppose that every irreducible component of $X^\mathrm{irr}(M)$ has dimension $1$. For a slope $\gamma\in H_1(\partial M;\mathbb{Z})$, let $\mathrm{tr}_\gamma:X^\mathrm{irr}(M)\to\mathbb{C}$ be the trace function, let $\mathrm{Tor}(M;\mathfrak{g}_\rho,\gamma)$ be the adjoint Reidemeister torsion, let $d_\gamma$ be the degree appearing in the paper's notation, and let $i_*:H_1(\partial M;\mathbb{Z})\to H_1(M;\mathbb{Z})$ be induced by inclusion. Twisted-index integrality conjecture. For every slope $\gamma$,

$$\sum_{[\rho]\in\mathrm{tr}_\gamma^{-1}(z)}\left(d_\gamma\cdot\mathrm{Tor}(M;\mathfrak{g}_\rho,\gamma)\right)^{g-1}\in\begin{cases}\mathbb{Z}[z]&\text{if }\gamma\in\operatorname{Ker}i_*\mathbb{Z}[z^2]&\text{otherwise}\end{cases}$$

for generic $z\in\mathbb{C}$. This prediction is derived from the finite-dimensionality and charge symmetries of the twisted-index Hilbert space; its general validity is not established in the source.