Tame purity conjecture for open de Rham spaces

Let $\mathscr{M}_{\bm \mu}^*$ be the tame open de Rham space associated with a generic tuple of semisimple adjoint orbits, and let $\mathscr{M}_{\mathbf{B}}^{{\bm \mu}, \mathbf{r}}$ and $\mathscr{M}_{{\bm \mu}, \mathbf{r}}^*$ denote the corresponding character variety and open de Rham space with the indicated additional irregularity data. The Riemann–Hilbert map induces

$$\nu_a^*: H^*(\mathscr{M}_{\mathbf{B}}^{{\bm \mu}, \mathbf{r}},\mathbb{Q})\to H^*(\mathscr{M}_{{\bm \mu}, \mathbf{r}}^*,\mathbb{Q}).$$

Purity conjecture. The map $\nu_a^*$ preserves mixed Hodge structures and is an isomorphism on the pure parts. This predicts that the pure cohomological information of the tame character variety agrees with that of the corresponding open de Rham space; the supplied text gives no resolution status.