Classicality detection conjecture via the Fontaine operator

Let $\mathbb{T}^S(K^p,\mathcal{O})_{\mathfrak{m}}$ be the localized Hecke algebra, let $\varphi:\mathbb{T}^S(K^p,\mathcal{O})_{\mathfrak{m}}\to\mathcal{O}$ be an eigensystem, and let $\rho_\varphi$ and $\chi_\varphi$ be the associated Galois characters or representations. Let $N$ be the first map of $ADR^{\Psi-\mathrm{la}}_{\lambda}$. Fontaine-operator detection conjecture. If $\rho_\varphi|_{G_{F_w}}$ is de Rham of $p$-adic Hodge type $\lambda_w$ for every $w\mid v$, if $\chi_\varphi|_{G_{F_{0,v^c}}}$ is de Rham of $p$-adic Hodge type $\lambda_0$, and if

$$\widehat{H}^d(S_{K^p},V_{\lambda^{\Psi}})_{\mathfrak{m}}^{\Psi-\mathrm{la}}[\varphi]\neq 0,$$

then

$$\mathrm{Ker}H^d(\mathscr{F}\!\ell,N)_{\mathfrak{m}}[\varphi]\neq 0.$$

This is intended to detect the relevant classical eigensystem in the kernel of the Fontaine-operator complex; the supplied text does not state a proof or resolution.