Resurgence and radial-limit conjecture for Habiro elements of knots

Let $K$ be a knot. Let $F_K(x)=\Phi_K(e^{-1/x})$ be the formal expansion of the Habiro element $\Phi_K$ around a root of unity, let $G_K(p)$ denote the Borel transform of $F_K(x)$, and let $S_K^{\mathrm{med}}(x)$ denote its median Borel sum. Write $A\stackrel{.}{=}B$ when $A$ and $B$ agree up to a prefactor depending on $K$. Resurgence and radial-limit conjecture. For every knot $K$, (1) $F_K(x)$ has a resurgent Borel transform $G_K(p)$; (2) $S_K^{\mathrm{med}}(x)$ is analytic on $\operatorname{Re}(x)>0$ and has radial limits at the points $\frac{1}{2\pi i}\mathbb{Q}$ of its natural boundary; and (3), if (2) holds, then for every $0\neq\alpha\in\mathbb{Q}$,

$$S_K^{\mathrm{med}}\left(-\frac{1}{2\pi i\alpha}\right):=\lim_{x\to-\frac{1}{2\pi i\alpha}}S_K^{\mathrm{med}}(x),\qquad \operatorname{Re}(x)>0,$$

satisfies

$$S_K^{\mathrm{med}}\left(-\frac{1}{2\pi i\alpha}\right)\stackrel{.}{=}\Phi_K(e^{2\pi i\alpha}).$$

This is the main conjecture attributed in the source to the cited work, within a broader program concerning analytic continuation of knot invariants arising from Chern–Simons theory. The supplied material gives no evidence that it has been resolved.