Stationary-phase conjecture for figure-eight knot surgeries

For an integer $p$, define

$$V_p(z,w):=-\operatorname{Li}_2(zw)+\operatorname{Li}_2\left(\frac{z}{w}\right)+\frac{p}{4}(\log z)^2-\log z\log w.$$

Let $M_p$ be the closed three-manifold obtained from the three-sphere by $p$-surgery along the figure-eight knot. Stationary-phase conjecture for figure-eight knot surgeries. There exists $(\zeta_p,\omega_p)$ such that

$$\frac{\partial V_p}{\partial z}(\zeta_p,\omega_p)=\frac{\partial V_p}{\partial w}(\zeta_p,\omega_p)=0,$$

and

$$V_p(\zeta_p,\omega_p)=\operatorname{CS}(M_p)+\sqrt{-1}\operatorname{Vol}(M_p).$$

This is presented as a weaker but more precise conjecture than the general closed-three-manifold volume conjecture, specializing the proposed asymptotics to integral surgeries on the figure-eight knot.