Curious Poincaré duality for wild character varieties

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 $WH({\mathcal{M}}_\B^{{\bm \mu},{\bf r}};q,t)$ be its mixed Hodge polynomial. Curious Poincaré duality conjecture.

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

The identity is deduced formally in the paper from the preceding mixed-Hodge-polynomial conjecture and a symmetry of the polynomials $\mathbb{H}_{{\bm \mu},{\bf r}}$. Thus it remains conjectural to the extent that the preceding conjecture remains open.