The weak Serre weight conjecture for globally defined weights

Let $G(k)$ be the finite reductive group in the local setup, let $\overline{\rho}_0$ be the local residual representation, and let $\overline{r}$ be its suitable globalization. Define $W_v(\overline{r})$ by

$$W_v(\overline{r}):= \{\sigma \mid \operatorname{Hom}_{\mathrm{K}}(\sigma,\pi(\overline{r})|_{\mathrm{K}})\neq 0\}.$$

Let $W^g(\overline{\rho}_0)$ be the set of Serre weights $\sigma$ of $G(k)$ such that $\overline{\rho}_0\in {\mathcal{C}}_\sigma(\mathbf{F})$. Weak Serre weight conjecture. One should have

$$W^g(\overline{\rho}_0)\subset W_v(\overline{r}).$$

This is a weak form of the expected Serre weight conjectures. The broader expectation is that the weights depend only on $\overline{\rho}_0$ and admit descriptions via inertia in generic tame cases or Breuil--Mészard cycles in general; the inclusion itself is presented as conjectural.