Let $G$ be a $p$-adic group, i.e., the group of $E$-points of some connected reductive group over a $p$-adic local field $E$. Let $\mathrm{Irr}(G)$ denote the set of equivalence classes of smooth irreducible representations of $G$. Assume that we are given some $\rho \in \mathrm{Irr}(G)$ such that, for some $i_0\geq 1$, we have $$\forall i\geq i_0, \forall \pi \in \mathrm{Irr}(G),\quad \mathrm{Ext}^i_G(\rho,\pi) = 0.$$
Does it follow that $\mathrm{Ext}^i_G(\rho,\pi) = 0$ for $i\geq i_0$ and for any smooth representation $\pi$ of $G$?
By induction, the statement holds if $\pi$ has finite length as a $G$-module. However, I fail to solve the case of more general $\pi$. Even the case of finitely generated $\pi$ seems unclear, and I can't produce a concrete counterexample either.