The optimistic semisimple Serre-weight conjecture

Let $\text{undefined}:\text{undefined}\to{}^LG(\text{undefined})$ be a semisimple $L$-parameter. Let $W^{\text{undefined}}(\text{undefined})$ be the set of Serre weights arising from crystalline lifts, and let $W^{\text{undefined}}(\text{undefined})$ be the set of Serre weights described by the explicit inertial condition: $F_{\text{undefined}}$ belongs to it when there are $\text{undefined}'\text{undefined}$ and $wW$ such that $\text{undefined}|_{I_{\text{undefined}}}\text{undefined}(\text{undefined}',w)$. Finally, let $W(\text{undefined})$ denote the conjectural true set of Serre weights. The optimistic semisimple Serre-weight conjecture. If $\text{undefined}$ is semisimple, then

$$W(\text{undefined})=W^{\text{undefined}}(\text{undefined})=W^{\text{undefined}}(\text{undefined}).$$

The conjecture identifies the true Serre weights with both the crystalline and explicit sets. The inclusion $W(\text{undefined})W^{\text{undefined}}(\text{undefined})$ is known in the usual global settings, while the asserted equality in general remains open.