Hausel–Rodriguez-Villegas genus-one combinatorial identity

For a partition $\lambda$, let $\mathcal{H}_{\lambda}(z,w)$ be the genus-one hook function, and let $lvertlambdarvert$ denote the size of $\lambda$. Hausel–Rodriguez-Villegas genus-one conjecture.

$$\sum_{\lambda} \mathcal{H}_{\lambda}\big(q^{1/2},t^{-1/2}\big) T^{\lvert\lambda\rvert}= \prod_{i,j,k\geqslant 1} \frac{(1-q^{i-1/2}t^{j-1/2}T^k)^2}{(1-q^{i-1}t^{j-1}T^k)(1-q^it^jT^k)}.$$

The source states that, in genus $1$, this identity is equivalent to the main mixed-Hodge-polynomial conjecture. The supplied text does not establish the identity, so its resolution is left open here.