Newton-stratum classicality conjecture for geometric de Rham cohomology

Let $G_{\mathbb{Q}_p}:=\operatorname{Res}_{F_w/\mathbb{Q}_p}\mathrm{GL}_{2,F_w}$, let $\mu$ be the specified minuscule cocharacter, and let $\mathscr{F}\!\ell^b$ be the Newton stratum indexed by $b\in B(G_{\mathbb{Q}_p},\mu^{-1})$. Write $j_b:\mathscr{F}\!\ell^b\hookrightarrow\mathscr{F}\!\ell$ for the natural immersion. Newton-stratum classicality conjecture. For every $b\in B(G_{\mathbb{Q}_p},\mu^{-1})$, the cohomology group

$$H^d(\mathscr{F}\!\ell,j_{b!}j_b^*GDR^{\mathrm{la}}_0)$$

has a generalized Hecke eigenspace decomposition by classical eigensystems. The source presents this as a more precise conjecture reducing the preceding classicality conjecture via excision; the supplied evidence leaves it open.