  In this article, we generalize Duflo's conjecture to understand the branching
laws of non-discrete series. We give a unified description on the geometric
side about the restriction of an irreducible unitary representation $\pi$ of
$\mathrm{GL}_n(k)$, $k=\mathbb{R}$ or $\mathbb{C}$, to the mirabolic subgroup,
where $\pi$ is attached to a certain kind of coadjoint orbit.
