ABPS twisted-extended-quotient compatibility conjecture

Let $(M',\xi_M,\mathcal T_M,\bar h_M)$ be a Levi subgroup of $(G',\xi,\mathcal T,\bar h)$, let $s=(M',\sigma)$ be a cuspidal datum for $G'$, and let $\hat s=({}^LM,\psi,v,\varrho)$ be a cuspidal datum for ${}^LG$ such that $(\psi,v,\varrho)$ is relevant for $(M',\xi_M,\mathcal T_M,\bar h_M)$. Suppose $G$ satisfies the reduction conjecture referred to as Theorem 55, and suppose $\dot{\mathfrak s}_{G'}$ corresponds to $\mathfrak s_{\widehat G}$ as in Theorem 56. ABPS compatibility conjecture. There is an isomorphism of twisted extended quotients fitting into the commutative diagram displayed in the source, with vertical maps given by projections to the naive quotients. This predicts that the Bernstein-block parameterizations on the representation and dual-group sides are identified by local Langlands reciprocity.