The genus-one partition identity for mixed Hodge polynomials

Let $a$ and $l$ denote the arm- and leg-lengths of a box in the Ferrers diagram of a partition $[200~\lambda$, and let $T,z,w$ be formal variables. The sum below is over all partitions, while the product is over all boxes in each Ferrers diagram.

Genus-one partition identity. The following combinatorial identity holds:

$$\sum_\lambda \prod \frac{\left(z^{2a+1}-w^{2l+1}\right)^2}{(z^{2a+2}-w^{2l})(z^{2a}-w^{2l+2})}\,T^{|\lambda|}=\exp\left(\sum_{k\geq 1}\frac{(z^k-w^k)^2}{(z^{2k}-1)(1-w^{2k})(1-T^k)}\frac{T^k}{k}\right).$$

This is an unproved identity related to Macdonald identities and the Weyl–Kac character formula; its representation-theoretic meaning is unclear. The analogous genus-zero identity is proved in the paper using a result of Garsia and Haiman.