Weissman's local Langlands correspondence for basic covers

Let $G$ be the underlying group, let $\textnormal{\textsf{Basic}}_G$ be the set of basic data, and for each $\beta\in\textnormal{\textsf{Basic}}_G$ let $\widetilde G_\beta$ be the corresponding cover. Let $\Pi(\widetilde G_\beta)$ denote its relevant representations, and let $\Phi(\widetilde H)$ be the set of isomorphism classes of semisimple L-parameters. Weissman's local Langlands correspondence for basic covers. For each $\beta\in\textnormal{\textsf{Basic}}_G$, there is a natural finite-to-one map

$$\operatorname{LLC}_\beta:\Pi(\widetilde G_\beta)\longrightarrow\Phi(\widetilde H).$$

This is the formulation of the same local Langlands correspondence simultaneously for the basic covers indexed by $\beta$. The source gives no evidence that it has been resolved.