Motivic Langlands generalization using the group $M_{K,2}$

Let $K$ be a finite extension of $\mathbb{Q}_p$ or $\mathbb{F}_q((z))$ with $\ell\neq p$ and $p>2$. The source considers motivic Schur-irreducible sheaves in the derived categories

$$D(\mathrm{Bundle}_{G,2})_{\mathrm{motivic},\blacksquare} \quad\text{and}\quad D(\mathrm{Bundle}_{G})_{\mathrm{motivic},\blacksquare},$$

with the latter related to smooth representations of $G(K)$ through the Bernstein center. Let $M_{K,2}$ be the fiber product of $M_K$ and $W_{K,2}$ over the relevant quotient. Motivic generalized Langlands conjecture. The motivic formalism of Scholze should admit a well-defined generalization to the present setting using the group $M_{K,2}$. The claim aims to extend the motivic geometric Langlands description, including specialization to $G(K)$-representations over $\overline{\mathbb{Q}}_\ell$, to the generalized group $M_{K,2}$; the source gives no proof or resolution, so it remains open.