Non-equivalence of pro-$p$-Iwahori invariants

Suppose $F$ is a nonarchimedean local field whose residue field is a proper extension of ${\mathbb{F}}_p$ with $p>3$. Generalizing results of Ghate–Le–Sheth, we show that any split, connected, reductive group $G$ over $F$ which is not a torus admits a smooth, irreducible, non-admissible mod $p$ representation. We use this to show that the functor of pro-$p$-Iwahori invariants does not induce an equivalence between the category of smooth mod $p$ $G$-representations generated by their pro-$p$-Iwahori invariant vectors and modules over the pro-$p$-Iwahori–Hecke algebra, contrary to what happens for ${\text{GL}}_2({\mathbb{Q}}_p)$.