Serre weight conjecture for $GSp_4$ over totally real fields

Let $F$ be a totally real field, let $p$ be a prime split completely in $F$, and let $\overline{\rho}:G_F\longrightarrow {\rm GSp}_4(\overline{\mathbb{F}}_p)$ be a mod $p$ irreducible automorphic Galois representation. Let $\mathcal{I}^{{\rm BC}}(\overline{\rho})$ be the collection of finite solvable extensions used for base change, and for any non-empty subset $\mathcal{I}\subset\mathcal{I}^{{\rm BC}}(\overline{\rho})$ define

$$W_\mathcal{I}(\overline{\rho})=\bigcup_{L\in \mathcal{I}}W(\overline{\rho}|_{G_L}).$$

For $\ast\in\{{\rm cris},{\rm pd-cris}\}$, let

$$W_{\ast,\mathcal{I}}(\overline{\rho})=\bigcup_{L\in \mathcal{I}}W_\ast(\overline{\rho}|_{G_L}),$$

where $W_{{\rm cris}}$ and $W_{{\rm pd-cris}}$ are the Serre weights arising respectively from crystalline and potentially diagonalizable crystalline lifts of the prescribed regular Hodge--Tate weights. Serre weight conjecture. For every non-empty subset $\mathcal{I}\subset\mathcal{I}^{{\rm BC}}(\overline{\rho})$,

$$W_\mathcal{I}(\overline{\rho})=W_{{\rm pd-cris},\mathcal{I}}(\overline{\rho})=W_{{\rm cris},\mathcal{I}}(\overline{\rho}).$$

This formulates Serre's weight conjecture for $GSp_4$ over totally real fields in terms of crystalline lifts, following the crystalline-lift philosophy for Serre's conjecture. The statement concerns the equality of automorphic Serre weights with those predicted by potentially diagonalizable crystalline lifts and by crystalline lifts after the permitted solvable base changes.