Weissman's local Langlands correspondence for a cover

Let $F$ be the local field and let $\widetilde G$ be the cover under consideration, with associated dual group data $\widetilde H$. Write $\Pi(\widetilde G)$ for the relevant set of genuine irreducible representations and let $\Phi(\widetilde H)$ denote the set of isomorphism classes of semisimple L-parameters, namely sections of $\widetilde H\to */W_F$. Weissman's local Langlands correspondence. There is a natural finite-to-one map

$$\operatorname{LLC}:\Pi(\widetilde G)\longrightarrow\Phi(\widetilde H).$$

This is Weissman's version of the local Langlands correspondence for covers, extending the expected parametrization of representations by L-parameters. The source gives no evidence that the conjecture has been resolved.