Finite-multiplicity extension of the p-reductivity conjecture

Let $\Omega$ be a bounded, strongly pseudoconvex domain in $\mathbb{C}^m$, and consider submodules of $B^2(\Omega)\otimes\mathbb{C}^k$. Finite-multiplicity p-reductivity conjecture. The same conclusion as the p-reductivity conjecture should hold for these submodules: when the orthogonal complement is spanned by joint generalized eigenvectors for the adjoints of the module action, both the submodule and its orthogonal complement should be $p$-reductive for $p>m$. This is stated as an expected finite-multiplicity generalization of the preceding conjecture and remains open.