Asymptotic formula for the meromorphic 3D-index

Let $M$ be a one-cusped hyperbolic 3-manifold, let $\hat M$ be its end compactification, and let $\mathcal{X}^{\mathrm{geo}}$ be the relevant set of conjugacy classes of geometric-obstruction representations. For a non-real conjugate pair $\{[\rho],[\overline\rho]\}$, let $z$ be an associated algebraic solution, $\tau(z)$ its amplitude, and $\operatorname{Vol}(\rho)=-\operatorname{Vol}(\overline\rho)$ its algebraic volume. Main conjecture. As $-1/\hbar=\kappa\to+\infty$,

$$\begin{aligned} \mathcal{I}_{M,(0,0)}(\hbar)&=|H_1(\hat M;\mathbb F_2)|\Biggl(\beta(M)\kappa+\sqrt{8\pi}\sum_{\{[\rho],[\overline\rho]\}\subset\mathcal{X}^{\mathrm{geo}}\setminus\mathcal{X}^{\mathrm{geo}}_\mathbb{R}}\tau(z)\sqrt{\kappa}\\ &\qquad\qquad\cdot\cos\Bigl(\kappa\operatorname{Vol}(\rho)+\tfrac{\pi}{4}(N_+-N_-+\Sigma(\omega))\Bigr)\Biggr)+o(1). \end{aligned}$$

Here $\beta(M)$ is the beta invariant, $N_+$ and $N_-$ count positively and negatively oriented tetrahedra, and $\Sigma(\omega)$ is the Hessian signature. The conjecture predicts the full oscillatory asymptotic expansion of the meromorphic 3D-index; the source presents it as the Main Conjecture and gives no resolution evidence.