Generic de Rham cohomology concentration conjecture

Let $\overline{\operatorname{Sh}}$ be the Siegel threefold or a unitary Shimura variety, and let $\mathfrak{m}\subseteq\mathbb{T}$ be generic and non-Eisenstein. Let $\lambda\in X^*(T)$ be generic, and suppose there exists $\mu\in X_1(T)$ such that $L(\lambda)$ is a Jordan--Hölder constituent of $V(\mu)_{\overline{\mathbb{F}}_p}$. Generic de Rham concentration conjecture. Then

H^\bullet_{\operatorname{dR}}(\overline{\operatorname{Sh}}^\operatorname{tor},\underline{L}(\lambda))_\mathfrak{m}

is concentrated in middle degree. Here genericity of $\lambda$ is understood in the sense specified in the cited reference. This concentration conjecture would imply the weak form of the de Rham weight conjecture once the corresponding generic middle-degree concentration for mod $p$ étale cohomology is known. It remains open in the stated generality.