Cohen–Macaulayness of the symplectic decomposition poset

Let ${\mathbb{K}}$ be any field and let $V$ be a finite-dimensional ${\mathbb{K}}$-vector space with a non-degenerate symplectic form $\Psi$. Let $\operatorname{\mathcal{S}}((V,\Psi),\mathbin{\perp})$ denote the poset of decompositions of $(V,\Psi)$ into mutually orthogonal nonzero symplectic subspaces.

Cohen–Macaulayness conjecture. The poset $\operatorname{\mathcal{S}}((V,\Psi),\mathbin{\perp})$ is Cohen–Macaulay.

The conjecture is needed to obtain Cohen–Macaulayness for the decomposition posets in the paper. It is posed for arbitrary fields; the preceding discussion notes that related results are known under restrictions on the field, while the general case remains open.