Gang–Kim–Yoon's vanishing conjecture for adjoint Reidemeister torsions

Let $M$ be a compact $3$-manifold with a torus boundary whose interior admits a hyperbolic structure. Suppose that the character variety of irreducible $\operatorname{SL}_2(\mathbb C)$-representations of $\pi_1(M)$ consists of only irreducible components of dimension $1$. For a slope $\gamma\in H_1(\partial M;\mathbb Z)$, let $\operatorname{tr}_\gamma$ denote the trace function of $\gamma$ on the character variety, and let $\mathbb T_{M,\gamma}$ be the adjoint Reidemeister torsion with respect to $\rho$ and $\gamma$. Gang–Kim–Yoon's conjecture. For generic $z\in\mathbb C$,

$$\sum_{[\rho]\in\operatorname{tr}_\gamma^{-1}(z)}\frac{1}{\mathbb T_{M,\gamma}}=0.$$

The conjecture is motivated by the 3D–3D correspondence and is supported by examples such as the figure-eight knot exterior, hyperbolic twist knot exteriors, and hyperbolic two-bridge knots. The paper provides infinitely many counterexamples, so the conjecture is refuted.