Classical eigensystem decomposition for the Fontaine-kernel cohomology

Let $\mathscr{F}\!\ell$ be the flag variety, let $N_0$ be the Fontaine operator, let $d$ be the relevant cohomological degree, and let $\mathfrak{m}$ be the specified Hecke ideal. Fontaine-kernel classicality conjecture. The kernel

$$\mathrm{Ker}H^d(\mathscr{F}\!\ell,N_0)_{\mathfrak{m}}$$

has a generalized eigenspace decomposition by classical eigensystems. This is proposed as the classicality statement needed to analyze the de Rham cohomology associated with the Fontaine operator; the supplied text does not establish it or give evidence resolving it.