On the relationship between distinction and irreducibility of parabolic induction

Let $U_{2n}$ denote the quasi-split unitary group over 2n variables with respect to a quadratic extension $E/F$ of p-adic fields. In this short note, we relate ${\mathrm{GL}}_n(F)$-distinction of ladder representations of ${\mathrm{GL}}_n(E)$ with irreducibility of its Siegel parabolic induction in $U_{2n}$.