Topological invariance of the Mellin--Barnes coefficient for real representations

Let $M$ be a connected, non-compact, orientable 3-manifold with one toroidal end and a complete hyperbolic structure, and let $\rho:\pi_1(M)\to PSL(2,\mathbb{R})$ be an irreducible boundary-parabolic representation with the geometric obstruction class. For an ideal triangulation $\mathcal{T}$ and the nonempty set $\Omega_\rho(\mathcal{T})$ of associated $\mathbb{Z}_2$-taut angle structures, let $I_{\mathcal{MB}}(\mathcal{T},\omega)$ denote the corresponding Mellin--Barnes contribution. Mellin--Barnes invariance conjecture. There exists a well-defined quantity $I_{\mathcal{MB}}(\rho)\in\mathbb{R}$, depending only on the topology of $M$ and the conjugacy class of $\rho$, such that

$$I_{\mathcal{MB}}(\rho)=\sum_{\omega\in\Omega_\rho(\mathcal{T})}I_{\mathcal{MB}}(\mathcal{T},\omega).$$

The conjecture is motivated by the invariance of the total beta invariant under Pachner moves and would make an individual Mellin--Barnes integral topological when only one taut angle structure occurs. Its general validity is left open.