Conjecture on p-reductivity of Bergman submodules and orthogonal complements

Let $\Omega$ be a bounded, strongly pseudoconvex domain in $\mathbb{C}^m$ with smooth boundary, let $B^2(\Omega)$ be its Bergman space, and let $\mathcal{S}\subseteq B^2(\Omega)$ be a submodule. Suppose that $\mathcal{S}^{\bot}$ is spanned by joint generalized eigenvectors for the adjoints of the operators defining the module action. The p-reductivity conjecture. Both $\mathcal{S}$ and $\mathcal{S}^{\bot}$ should be $p$-reductive for $p>m$, meaning that their module cross-commutators belong to the Schatten class $\mathcal{S}_p$. The claim is motivated by submodules associated with varieties, ideals, and higher-order vanishing, but the source presents it as an optimistic conjecture with scant evidence.