Toroidal volume conjecture for hyperbolic links

Let $T^2\times I$ be the thickened torus, let $L\subset T^2\times I$ be a link, and let $J_n^T(L;e^{2\pi i/n})$ denote the $n$th toroidal colored Jones polynomial evaluated at $e^{2\pi i/n}$. Assume that $(T^2\times I)\setminus L$ is hyperbolic.

Toroidal volume conjecture.

$$\lim_{n\to\infty}\frac{2\pi}{n}\log\left|J_n^T\left(L;e^{2\pi i/n}\right)\right|=\operatorname{Vol}\bigl((T^2\times I)\setminus L\bigr).$$

The paper presents this as the volume-conjecture analogue for links in the thickened torus; it proves the claim for the 2-by-2 square weave and gives computational evidence for other links.