The generating-series conjecture for perverse numbers of parabolic Hitchin systems

Let $\bm{\mu}$ be the multipartition encoding the parabolic data of $M_B(n)$, let $k$ be as defined in the paper, and let $\mathbb{H}_{\bm{mu}}$ be the function defined in Section 1.1 of the cited work. The generating-series conjecture. For the $\widetilde{A_0}(n)$ case, one has

$$\prod_{m=1}^\infty\frac{(1+s^mq^{m}t^{2m-1})^2}{(1-s^mq^{m-1}t^{2m-2})(1-s^mq^{m+1}t^{2m})}=\sum_{n=0}^\infty (st^2q)^n\mathbb{H}_{\bm{\mu}}(-\sqrt{q},\frac{\sqrt{q}}{t}).$$

For the other four cases, one has

$$\prod_{m=1}^\infty\frac{1}{(1-s^mq^{m-1}t^{2m-2})(1-s^mq^mt^{2m})^k(1-s^mq^{m+1}t^{2m})}=\sum_{n=0}^\infty (st^2q)^n\mathbb{H}_{\bm{\mu}}(-\sqrt{q},\frac{\sqrt{q}}{t}).$$

These identities give the proposed equality between the computed perverse-number generating series and the mixed-Hodge-number generating series predicted for the corresponding character varieties.