The critical-surface asymptotic conjecture for quantum invariants of figure-8 knot surgeries

Let $p/q$ be a rational surgery coefficient, let $d$ be the inverse of $p$ modulo $q$, let $(a,b)\in\{0,1\}^2$, $n\in\mathbb{Z}/|q|\mathbb{Z}$, and $(\mu,\nu)\in\{\pm1\}^2$. Define

$$\Phi_n(x,y)=-\frac{dn^2}{q}-\frac{p}{4q}x^2+\frac{n}{q}x-xy+\frac{1}{4\pi^2}\left(\operatorname{Li}_2(e^{2\pi i(x+y)})-\operatorname{Li}_2(e^{2\pi i(x-y)})\right),$$ $$\Phi_n^{a,b}(x,y)=a(x+y)+b(x-y)+\Phi_n(x,y),$$

and let ${\mathcal S}=\{(x,y)\in\mathbb{R}\times\mathbb{C}\mid e^{2\pi i y}\in(-\infty,0)\}$. Critical-surface asymptotic conjecture. There exist surfaces $\tilde{\Sigma}_{a,b}^{\mu,\nu,n}\subset\mathbb{C}^2$ such that

$$\bar{\tau}_r(M_{p/q})\sim Cr\sum_{n\in\mathbb{Z}/|q|\mathbb{Z}}\sum_{(a,b)\in\{0,1\}^2}\sum_{(\mu,\nu)\in\{\pm1\}^2}\mu\nu\int_{\tilde{\Sigma}_{a,b}^{\mu,\nu,n}}\tilde g_n(x)e^{2\pi i r\Phi_n^{a,b}(x,y)}\,dx\,dy,$$

where $C$ depends only on $p$ and $q$, and $\tilde g_n$ is an $r$-independent function of $x\in\mathbb{C}$. If $p/q\ne0$, the surfaces can be chosen so that every critical point of $\Phi_n^{a,b}$ lying on the corresponding surface belongs to ${\mathcal S}$. This is a proposed integral representation for the leading asymptotics of the quantum invariant; the paper presents it as an analog of the figure-8 colored Jones analysis, with only partial analytic support given.