The transfer conjecture for tame blocks of p-adic linear groups

Let $F$ be a nonarchimedean local field, let $\mathbf{G}$ and $\mathbf{G'}$ be connected reductive groups of ${\rm GL}$-type over $F$, and let $\xi:{}^{L}\mathbf{G'}\rightarrow{}^{L}\mathbf{G}$ be an $L$-homomorphism. Fix an admissible parameter $\phi':I_F^{(\ell)}\rightarrow{}^{L}\mathbf{G'}$ and set $\phi=\xi\circ\phi'$. Write $C_{\hat{\mathbf G}}(\phi)$ for the centralizer of the image of $\phi$.

The transfer conjecture. Suppose that $\xi$ induces an isomorphism

$$C_{\hat{\mathbf G'}}(\phi')\simeq C_{\hat{\mathbf G}}(\phi)$$

and that the projection of $\xi(W_F)$ to $\hat G(\overline{\mathbb Q}_\ell)$ is bounded. Then there is an equivalence of categories

$$\operatorname{Rep}_{\phi'}(G')\simeq\operatorname{Rep}_{\phi}(G)$$

that interpolates the Langlands transfer $\xi_*$ on irreducible $\overline{\mathbb Q}_\ell$-representations. This conjectural equivalence would provide a categorical Langlands transfer between the corresponding blocks, extending the known transfer on irreducible characteristic-zero representations; its general validity is not established in the supplied source.