Functoriality conjecture for integral blocks of p-adic groups

Let $\mathbf{G}'$ and $\mathbf{G}$ be groups of ${\rm GL}$-type with an $L$-homomorphism $\xi:{^{L}\mathbf{G}'}\rightarrow {^{L}\mathbf{G}}$. Let $\phi':I_F^{(\ell)}\rightarrow {^{L}\mathbf{G}'}$ be an admissible parameter and set $\phi=\xi\circ\phi'$. Denote by $C_{\hat{\mathbf G}'}(\phi')$ and $C_{\hat{\mathbf G}}(\phi)$ the corresponding centralizers, and by $\operatorname{Rep}_{\phi',\mathbb Z_\ell}(G')$ and $\operatorname{Rep}_{\phi,\mathbb Z_\ell}(G)$ the associated blocks. Functoriality conjecture for integral blocks. If $\xi$ induces an isomorphism $C_{\hat{\mathbf G}'}(\phi')\simeq C_{\hat{\mathbf G}}(\phi)$ and the projection of $\xi(W_F)$ to $\hat G(\mathbb Q_\ell)$ is bounded, then there is an equivalence of categories

$$\operatorname{Rep}_{\phi',\mathbb Z_\ell}(G')\simeq \operatorname{Rep}_{\phi,\mathbb Z_\ell}(G)$$

that interpolates the Langlands transfer $\xi_*$ on irreducible $\mathbb Q_\ell$-representations. This is a proposed integral refinement of functorial transfer for blocks; the paper notes that the theorem is not proved in general and indicates that specific cases, such as unramified automorphic induction and totally ramified base change, can be established using results from Deligne–Lusztig theory.