The vector-valued generalization of the Fourier-coefficient cuspidality theorem

Let $\rho$ be a representation and let $\widetilde{M}_n(k,\rho,\widetilde{\Gamma})$ denote the corresponding space of vector-valued Siegel modular forms. Vector-valued generalization conjecture. The analogue of the paper's main theorem holds for $\widetilde{M}_n(k,\rho,\widetilde{\Gamma})$ for any given $\rho$. The paper notes that the scalar-valued or metaplectic cases are accessible, whereas the assertion is not clear for a general representation $\rho$; it therefore remains open.