Irregular purity conjecture for wild character varieties

Let $\mathscr{M}_{{\bm \mu},\mathbf r}^*$ be the open de Rham space of meromorphic connections with the prescribed regular and higher-order poles, and let $\mathscr{M}_{\mathbf B}^{{\bm\mu},\mathbf r}$ be the corresponding wild character variety. The irregular Riemann–Hilbert map is

$$\nu:\mathscr{M}_{{\bm \mu},\mathbf r}^*\to\mathscr{M}_{\mathbf B}^{{\bm\mu},\mathbf r}.$$

Irregular purity conjecture. The map $\nu$ induces

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

which preserves mixed Hodge structures and is an isomorphism on the pure parts. This predicts purity compatibility between the wild character variety and the de Rham moduli space; the supplied text gives no resolution status.