Vanishing identity for adjoint Reidemeister torsions of closed 3-manifolds

Let $\mathfrak{g}$ be the Lie algebra of a semisimple complex Lie group $G$, let $M$ be a connected closed oriented $3$-manifold, and let $R^{\rm irr}_G(M)$ be the set of conjugacy classes of irreducible representations $\pi_1(M)\to G$. For a representation class $\varphi$, write $\tau_\varphi(M)$ for its adjoint Reidemeister torsion. Assume that $R^{\rm irr}_G(M)$ is finite, and let $n\in\mathbb{Z}$ with $n\geq -1$. Vanishing identity. The sum

$$\sum_{\varphi\in R^{\rm irr}_G(M)}\bigl(2\tau_\varphi(M)\bigr)^n$$

lies in $\mathbb{Z}$. Furthermore, if $G=\operatorname{SL}_2(\mathbb{C})$, $M$ is hyperbolic, and $n=-1$, then the sum is zero. The identity is a proposed extension of the torsion integrality and vanishing phenomena for 3-manifolds with torus boundary; the paper studies cases supporting it, while the general assertion remains open.