The octahedron conjecture for the perverse-Hodge octahedron

Let $X$ be a compact hyperkähler manifold, and define

$$h^{i,k,d} \coloneqq \dim\left(\left(\operatorname{Gr}^{P}_{d+k}H^{2d}(X,\mathbb{C})\right)^{d+i,d-i}\right).$$

The perverse-Hodge octahedron conjecture. The convex hull of the non-zero $h^{i,k,d}$ is an octahedron. This is stronger than the known containment in a rhombic dodecahedron and is equivalent to a maximal-nilpotency statement for the operator $[L_{\beta},\Lambda_{\overline{\sigma}}]$ on $H^{2d}$; it remains open in general.