Reduction of integral blocks to unipotent blocks

Let $\mathbf G$ be a group of ${\rm GL}$-type, let $\phi:I_F^{(\ell)}\rightarrow {^{L}\mathbf G}$ be an admissible parameter, and choose an extension $\varphi:W_F\rightarrow {^{L}\mathbf G}$. Let $\mathbf G_\phi$ be the associated group of ${\rm GL}$-type with dual group $\hat{\mathbf G}_\phi=C_{\hat{\mathbf G}}(\phi)$, and let $\xi_\varphi:{^{L}\mathbf G}_\phi\hookrightarrow {^{L}\mathbf G}$ be the indicated embedding. Write $\operatorname{Rep}_{1,\mathbb Z_\ell}(G_\phi)$ for the principal integral block and $\operatorname{Rep}_{\phi,\mathbb Z_\ell}(G)$ for the block attached to $\phi$. Reduction conjecture for integral blocks. There is an equivalence of categories

$$\operatorname{Rep}_{1,\mathbb Z_\ell}(G_\phi)\simeq \operatorname{Rep}_{\phi,\mathbb Z_\ell}(G)$$

that extends the transfer of irreducible $\mathbb Q_\ell$-representations associated to $\xi_\varphi$. This conjecture reduces the study of arbitrary integral blocks to principal, or unipotent, integral blocks and is presented as a consequence of the preceding functoriality conjecture.