Homogeneity and indecomposability conjecture for mirabolic restrictions

Let $\pi$ be an irreducible representation of $\mathrm{GL}_n$, and let $M_n$ be the mirabolic subgroup. Use the paper's definition of a homogeneous representation and write $\pi|_{M_n}$ for the mirabolic restriction. Homogeneity and indecomposability conjecture. The representation $\pi$ is homogeneous and $\pi|_{M_n}$ is indecomposable. This extends the result proved in the paper for irreducible generic representations. The conjecture is also motivated by the expected unique irreducible submodule of the highest derivative; its validity for general irreducible representations remains open.