Mixed-Poincare conjecture for non-generic character stacks

Let $\mathcal{C}$ be a $k$-tuple of semisimple orbits, let $\alpha_{\mathcal{C}}$ be its associated dimension vector, and let $\mathcal{H}^*_{\gamma_{\mathcal{C}},\alpha_{\mathcal{C}}}$ be the specified set of dimension vectors. For each $\beta$ in this set, write $\mathbb{H}_{\beta}(z,w)$ for the associated rational function, and let $\operatorname{Coeff}_{\alpha_{\mathcal{C}}}$ and $\operatorname{Exp}$ denote coefficient extraction and plethystic exponential, respectively. Mixed-Poincare conjecture for non-generic character stacks. For any $k$-tuple of semisimple orbits $\mathcal{C}$,

$$\operatorname{Coeff}_{\alpha_{\mathcal{C}}}\left(\operatorname{Exp}\left(\sum_{\substack{\beta \in \mathcal{H}^*_{\gamma_{\mathcal{C}},\alpha_{\mathcal{C}}}}} \dfrac{(qt^2)\mathbb{H}_{\beta}\left(t\sqrt{q},\frac{1}{\sqrt{q}}\right)}{qt^2-1}y^{\beta}\right)\right)=\dfrac{H_c(\mathcal{M}_{\mathcal{C}},q,-t)}{(qt^2)^{-(\alpha_{\mathcal{C}},\alpha_{\mathcal{C}})}}.$$

This conjecture generalizes the generic mixed-Poincare formula and the paper’s proven E-series formula from generic to arbitrary semisimple monodromies; its validity is proposed but not established in the supplied text.