Generalized Langlands geometric objects for two-fold Weil coverings

Let $K$ be a finite extension of $\mathbb{Q}_p$ or $\mathbb{F}_q((z))$ with $\ell\neq p$ and $p>2$, let $G/K$ be reductive, and let $W_{K,2}$ denote the indicated two-fold covering group of the Weil group $W_K$. Write $G^\text{duali\'e}$ for the reductive dual group and

$$G^\text{Lan}=G^\text{duali\'e}\overset{\mathrm{right}}{\times}W_K.$$

For any condensed morphism from $W_{K,2}$ to $G^\text{Lan}$, the generalized Langlands conjecture. There should be geometric objects on the other side of the Langlands correspondence that functorially extend the usual Schur-irreducible sheaves over the moduli $v$-stack of vector bundles on Fargues–Fontaine curves. This proposes an extension of the usual geometric Langlands objects to the setting with the additional two-fold Weil-group information; the source does not establish existence or resolution, so the conjecture remains open.