Quasi-free Hilbert-module p-reductivity conjecture

Let $\mathcal{M}$ be a finite-rank, quasi-free, $p$-reductive Hilbert module over $A(\Omega)$, and let $\mathcal{S}$ be a submodule such that $\mathcal{S}^{\bot}$ is spanned by generalized eigenvectors for the adjoints of the operators defining the module action. Quasi-free p-reductivity conjecture. Then $\mathcal{S}$ and $\mathcal{S}^{\bot}$ should be $p$-reductive. The source describes this as a stronger possible result useful for the subsequent discussion; no proof or resolution is given.