The relative category equality for distinguished conjugacy classes in symplectic groups

Let $\operatorname{Sp}(n)$ be the compact symplectic group, let $\mathcal O_k$ be the distinguished conjugacy class associated with the $k$-th vertex of the fundamental alcove, and let

$$d_k=\min\{k,n-k\}.$$

The class $\mathcal O_k$ is identified with the quaternionic Grassmannian $\operatorname{Gr}_{d_k}(\mathbb H^n)$, and $\operatorname{cat}_{\operatorname{Sp}(n)}(\mathcal O_k)$ denotes its relative L-S category in $\operatorname{Sp}(n)$. The relative category equality. For $1\leq k\leq n-1$, one has

$$\operatorname{cat}_{\operatorname{Sp}(n)}(\mathcal O_k)=d_k.$$

The supplied context presents this as a conjecture-environment candidate, but the excerpt gives no surrounding assertion or status evidence establishing whether it is intended as an open conjecture, a proposed strengthening, or a result; consequently its resolution should be checked against the full paper.