Classicality conjecture for geometric de Rham complex cohomology

Let $\mathscr{F}\!\ell$ be the flag variety, let $GDR^{\mathrm{la}}_0$ be the geometric locally analytic de Rham complex, let $d$ be the relevant cohomological degree, and let $\mathfrak{m}$ be the specified Hecke ideal. Geometric de Rham classicality conjecture. The cohomology group

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

has a generalized Hecke eigenspace decomposition by classical eigensystems. This conjecture is proposed as a sufficient ingredient for the classicality argument; the supplied status evidence says that the broader classicality result depends on classicality of a rigid cohomology group that remains open.