Hilbert-scheme identity for character varieties

Let $\mathcal H_\lambda(z,w)$ be the partition function appearing in the source, let $\phi(0):=0$, and for every nonzero partition $\lambda$ define

$$\phi_\lambda(z,w):=\sum_{(i,j)\in\lambda}z^{j-1}w^{i-1},$$

where the sum is over the boxes of $\lambda$. Hilbert-scheme identity. The following formal power-series identity is conjectured:

$$1+(z^2-1)(1-w^2)\frac{\sum_\lambda\mathcal H_\lambda(z,w)\phi_\lambda(z^2,w^2)T^{|\lambda|}}{\sum_\lambda\mathcal H_\lambda(z,w)T^{|\lambda|}}=\prod_{n\geq1}\frac{(1-zwT^n)^2}{(1-z^2T^n)(1-w^2T^n)}.$$

It arises by combining the Hilbert-scheme formula with the main conjecture in the case $g=1$ and $\mu=(n-1,1)$, giving a purely combinatorial reformulation.