The link-level root-of-unity relation for CGP invariants

Let $L$ be a framed link in $S^3$ with $V$ components, let $B$ be its linking matrix, and let $L_{\alpha}$ denote the link with labels $\vec\alpha=(\alpha_1,\ldots,\alpha_V)\in\mathbb C^V$. Let $F_L$ be a nonzero formal power series in the stated coefficient module, and let $F_L^t$ be obtained by replacing each $x_I^n$ and $x_I^{-n}$ with the corresponding $tq^{\pm\alpha_I}$. The link-level root-of-unity conjecture. For every $r\ge2$ and every $\vec\alpha\in\mathbb C^V$,

$$\lim_{t\to 1}\lim_{q\to\exp(\frac{2\pi i}{r})} \frac{F_L^t(q^{\alpha_1},\dots,q^{\alpha_V},q)}{F_L^t(q^{r\alpha_1},\dots,q^{r\alpha_V},q^r)} =\mathrm N_r(S^3,L_{\alpha})q^{-\frac12(\vec\alpha^{T}B\vec\alpha-(r-1)^2\operatorname{Tr}B)}.$$

Although the radius of convergence may be less than one, the root-of-unity limit is conjectured to be rational in $t$, so that the limit $t\to1$ is interpreted by analytic continuation. This generalizes the knot-level relation to links and connects formal BPS series with CGP link invariants.