Weight-$\lambda$ Newton-stratum classicality conjecture

Let $G$ be the group in the source, let $\mu$ be the specified minuscule cocharacter, let $GDR^{\Psi-\mathrm{la}}_{\lambda}$ be the geometric locally analytic de Rham complex of weight $\lambda$, and let $j_b:\mathscr{F}\!\ell^b\hookrightarrow\mathscr{F}\!\ell$ be the natural immersion for $b\in B(G,\mu^{-1})$. Weight-$\lambda$ classicality conjecture. For every $b\in B(G,\mu^{-1})$, the group

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

has a generalized Hecke eigenspace decomposition by classical eigensystems of weight $\lambda$. Consequently,

$$H^d(\mathscr{F}\!\ell,GDR^{\Psi-\mathrm{la}}_{\lambda})$$

has such a decomposition as well. This is a proposed classicality statement for the Newton-stratum pieces and their global geometric de Rham cohomology; no resolution is given in the supplied text.