The de Rham weight part of Serre's conjecture for Shimura varieties

Let $(G,X)$ be a Shimura datum of Hodge type. Let $\mathbb{T}$ be the spherical Hecke algebra and let $\mathfrak{m}\subseteq\mathbb{T}$ be a maximal ideal. For an algebraic representation $V$ of $G$ over $\overline{\mathbb{F}}_p$, write $\underline{V}$ for the associated vector bundle with flat connection on the toroidal compactification \overline{\operatorname{Sh}}^\operatorname{tor}, and define

W_{\operatorname{dR}}(\mathfrak{m})=\{\lambda\in X_1(T):H^\bullet_{\operatorname{dR}}(\overline{\operatorname{Sh}}^\operatorname{tor},\underline{L(\lambda)})_\mathfrak{m}\neq 0\}.

Let $W(\mathfrak{m})$ denote the usual set of Serre weights, with the equality below interpreted via the surjection $\lambda\in X_1(T)\mapsto F(\lambda)$. The de Rham weight conjecture. The strong form asserts

$$W_{\operatorname{dR}}(\mathfrak{m})=W(\mathfrak{m}).$$

The weak form asserts that if $\operatorname{Sh}$ is a unitary Shimura variety or the Siegel threefold and $\mathfrak{m}$ is generic non-Eisenstein, then there exists $\epsilon\geq 0$, depending only on $G_{\mathbb{Q}}$, such that

$$W^\epsilon_{\operatorname{dR}}(\mathfrak{m})=W^\epsilon(\mathfrak{m}).$$

The conjecture gives a de Rham formulation of the weight part of Serre's conjecture. The weak form is expected to follow from generic concentration results for étale and de Rham cohomology, while the strong form remains conjectural in general.