Translation conjecture for coherent pairs and restriction cohomology

Let $(pi,pi')$ be a coherent pair with Harish-Chandra parameters $\lambda$ and $\lambda'$ sufficiently far from the walls. Let $(\sigma,\sigma')$ be an elementary pair, let $F$ and $F'$ be irreducible finite-dimensional representations of $G$ and $G'$, respectively, and let $A$, $A'$, $B$, $B'$, $T$, and $S$ be the morphisms described above. Translation conjecture. These data can be chosen so that the symmetry-breaking relation

$$(S \otimes \operatorname{pr}_{F})\circ A=A'\circ T$$

commutes and all induced morphisms in cohomology are isomorphisms. In particular, all six cohomology groups in the displayed diagram are one-dimensional in degree $q=q(\pi)=q(\pi')$. This predicts that the composite map from $H^{\ast}({\mathfrak{P}},K;\pi\otimes W_{\pi}^{\vee})$ to $H^{\ast}({\mathfrak{P}}',K';\pi'\otimes {W'_{\pi'}}^{\vee})$ obtained from $B_*$, translation, and $B'_*$ is an isomorphism for suitable data.