Parametrized asymptotic expansion conjecture for colored Jones polynomials

Let $K$ be a hyperbolic knot and let $M,N$ be integers. Set

$$s=\lim_{N\to\infty}\frac{M}{N+1/2}.$$

There should be a small neighborhood $U\subset\mathbb R$ of $s=1$ such that, for $s\in U$, there is a holomorphic function $\Phi_M^{(s)}(z)$ with the following properties: $\Phi_N^{(1)}(z)\to\Phi_0^{(1)}(z)$; the latter has a non-degenerate saddle point $z_0^{(1)}$ giving the complex volume, and there are smoothly varying non-degenerate saddle points $z_M^{(s)}$ satisfying the saddle-point equations and converging to $z_0^{(1)}$ as $M\to N$, with

$$\Phi_M^{(s)}(z_M^{(s)})\to\Phi_0^{(1)}(z_0)=\operatorname{Vol}(K)+i\operatorname{CS}(K).$$

Parametrized asymptotic expansion conjecture. The family determines the asymptotic expansion

$$J_M\left(K,e^{\frac{2\pi i}{N+1/2}}\right)\sim \text{(constant)}\times\left(\frac{N}{2\pi i}\right)^{3/2}\frac{\exp\left(\frac{N+1/2}{2\pi}\Phi_M^{(s)}(z_M^{(s)})\right)}{\sqrt{\frac{d^2}{dz^2}\Phi_M^{(s)}(z_M^{(s)})}}.$$

This conjecture proposes a one-parameter saddle-point description of colored Jones asymptotics near the complete hyperbolic structure. In the supplied text it is presented as a conjectural framework, with the figure-eight case providing the motivating evidence.