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

Let $M$ be an oriented compact hyperbolic $3$-manifold with a torus boundary. A norm curve is a one-dimensional component of the character variety on which every trace function is non-constant; write $X^{\mathrm{nor}}(M)$ for the set of all norm curves. For a slope $\gamma\in H_1(\partial M,\mathbb Z)$, let $\operatorname{tr}\rho(\gamma)$ be the trace of $\rho(\gamma)$ and let $\mathbb T_{M,\gamma}$ be the adjoint Reidemeister torsion. Modified Gang–Kim–Yoon conjecture. For generic $z\in\mathbb C$,

$$\sum_{\substack{[\rho]\in X^{\mathrm{nor}}(M)\\ \operatorname{tr}\rho(\gamma)=z}}\frac{1}{\mathbb T_{M,\gamma}}=0.$$

The modification excludes extra character-variety components containing constant trace functions. The paper shows that this identity holds for all hyperbolic once-punctured torus bundles with tunnel number one, although its validity in the stated generality remains open.