The genus-one product identity for the character-variety generating function

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

Genus-one product identity.

$$\sum_\lambda\prod\frac{\left(z^{2a+1}-w^{2l+1}\right)^2}{(z^{2a+2}-w^{2l})(z^{2a}-w^{2l+2})}T^{|\lambda|}=\prod_{n\geq1}\prod_{r>0}\prod_{s\geq0}\frac{(1-z^{2s+1}w^{-2r+1}T^n)^2}{(1-z^{2s}w^{-2r+2}T^n)(1-z^{2s+2}w^{-2r}T^n)}.$$

This identity is a consequence of the main conjecture in the genus-one case, where the character variety is a two-dimensional torus; it is not presented as independently proved in the supplied text.