Homological branching laws for $p$-adic and mod $p$ representations of ${\rm GL}_2$ over a $p$-adic field

Let $F$ be a finite extension of $\mathbb Q_p$ and $p\geq 5$. In this paper, we establish homological branching laws for mod $p$ representations of $\mathrm{GL}_2(F)$ restricted to split and nonsplit maximal tori, treating principal and special series and, when $F=\mathbb Q_p$, supersingular represe