The first pro-$p$-Iwahori cohomology of mod-$p$ principal series for $p$-adic ${\text{GL}}_n$

Let $p\ge 3$ be a prime number and $F$ a $p$-adic field. Let $I_1$ denote the pro-$p$-Iwahori subgroup of ${\text{GL}}_n(F)$, and $\mathcal{H}$ the pro-$p$-Iwahori–Hecke algebra of ${\text{GL}}_n(F)$ with respect to $I_1$ (over a coefficient field of characteristic $p$). We compute the structure of ${\text{H}}^1(I_1,\pi )$ as an $\mathcal{H}$-module, where $\pi$ is a mod-$p$ principal series representation of ${\text{GL}}_n(F)$. We also give some partial results about the structure of ${\text{H}}^1(I_1,\pi )$ for a general split reductive group with irreducible root system.