Highest-weight compatibility conjecture for the Heckman–Opdam equivalence

Let $c$ be a parameter for which the Heckman–Opdam functor $F$ is an equivalence between the highest weight categories $\mathcal{O}_c$ and $\mathcal{O}_{\sigma(c)}$. Let $\Delta_c(E)$ denote the standard object indexed by $E$, let $s(c)$ be the shifted parameter appearing in the source, and let $\kappa_\sigma(E)$ be the corresponding relabeling of the index. Highest-weight compatibility conjecture. If $F$ is an equivalence, then it is an equivalence of highest weight categories and

$$F(\Delta_c(E)) \cong \Delta_{s(c)}(\kappa_\sigma(E)).$$

This is proposed as the final ingredient in the approach described; the supplied text gives no proof or resolution.