Wild character-variety mixed Hodge polynomial conjecture

Let ${\mathcal{M}}_\B^{{\bm \mu},{\bf r}}$ be the generic wild character variety, let $d_{{\bm \mu},{\bf r}}$ be its dimension, and let $\tilde{{\bm \mu}}=(\mu^1,\dots,\mu^k,(1^n),\dots,(1^n))$ with $r=r_1+\cdots+r_m$. Write $WH({\mathcal{M}}_\B^{{\bm \mu},{\bf r}};q,t)$ for its mixed Hodge polynomial. Wild character-variety conjecture. We have

$$WH({\mathcal{M}}_\B^{{\bm \mu},{\bf r}};q,t)=(qt^2)^{d_{{\bm \mu},{\bf r}}}\mathbb{H}_{\tilde{{\bm \mu}},r}(q^{-1/2},-tq^{-1/2}).$$

The paper proves the $t=-1$ specialization, so the conjecture extends that theorem to the full mixed Hodge polynomial. The source describes this as a conjectural partial answer to a problem concerning character varieties with matching mixed Hodge polynomials.