Reduction of level-zero blocks to principal blocks

Let $\psi:P_F\rightarrow {^{L}\mathbf G}$ be a wild parameter, and let $\mathbf G_\psi$ be the associated tamely ramified group of ${\rm GL}$-type with $\hat{\mathbf G}_\psi=C_{\hat{\mathbf G}}(\psi)$. Let $\xi:{^{L}\mathbf G}_\psi\hookrightarrow {^{L}\mathbf G}$ be the associated embedding, let $\operatorname{Rep}_{1}(G_\psi)$ be the principal block, and let $\operatorname{Rep}_{\psi}(G)$ be the direct factor attached to $\psi$. Wild-parameter reduction conjecture. There is an equivalence of categories

$$\operatorname{Rep}_{1}(G_\psi)\simeq \operatorname{Rep}_{\psi}(G)$$

that extends the transfer associated to the embedding $\xi$. This is proposed as a consequence of the integral-block reduction conjecture and would reduce the general level-zero category to a principal block for a tamely ramified group of ${\rm GL}$-type.