The isotropicity conjecture for the five canonical planes in the Debarre–Voisin tangent space

Let $Z_{\alpha}$ be the Debarre–Voisin variety associated with a general $3$-form, and let $\mathcal{T}_{Z_{\alpha}}$ denote its tangent space at the point under consideration. The five canonically defined vector spaces of dimension $2$ are subspaces of this tangent space.

Isotropicity conjecture. The five vector spaces of dimension $2$ canonically defined in the relevant construction are maximal isotropic subspaces for the symplectic form on $\mathcal{T}_{Z_{\alpha}}$ constructed by Debarre and Voisin.

This conjecture predicts that the five distinguished $2$-dimensional spaces form maximal isotropic subspaces for the natural symplectic structure. The supplied text gives heuristic evidence from computations and secant-line considerations, but does not state a resolution.