Chern-Simons reciprocity for closed 3-manifolds with finite irreducible character variety

Let $M$ be an oriented closed $3$-manifold, let $G$ be a simply connected complex simple Lie group, and let $\mathrm{R}^{\rm irr}_G(M)$ be the character variety of irreducible representations $\pi_1(M)\to G$. Write $\pi_0(\mathrm{R}^{\rm irr}_G(M))$ for its finite set of connected components, and let $\mathrm{CS}(\rho)\in\mathbb{C}/\mathbb{Z}$ be the Chern-Simons invariant. Chern-Simons reciprocity conjecture for $\mathrm{SL}_2(\mathbb{C})$. If $G=\mathrm{SL}_2(\mathbb{C})$ and $|\mathrm{R}^{\rm irr}_G(M)|<\infty$, then $c_G=24$ and

$$24\sum_{[\rho]\in\pi_0(\mathrm{R}^{\rm irr}_G(M))}\mathrm{CS}(\rho)=0\quad\text{in }\mathbb{C}/\mathbb{Z}.$$

This is the paper's stated result in the $\mathrm{SL}_2(\mathbb{C})$ case, while the preceding problem asks for a general integer $c_G$ and a topological interpretation of the resulting residue. The supplied status is unknown, so the database records it as open.