On freeness of compactly induced mod-$p$ representations of ${\mathrm{S}\mathrm{L}}_2(F)$

Let $p$ be a prime, and $F$ a non-archimedean local field with residue characteristic $p$ and ring of integers ${\mathcal{O}}_F$. Set $G_S:={\mathrm{S}\mathrm{L}}_2(F)$and $K_0:={\mathrm{S}\mathrm{L}}_2({\mathcal{O}}_F)$ . For a smooth irreducible ${\bar{\mathbb{F}}}_p$-representation $\sigma$ of $K_0$, we study the structure of the compact induction ${\mathrm{i}\mathrm{n}\mathrm{d}}_{K_0}^{G_S}(\sigma )$ as a left module over the standard spherical Hecke algebra ${\mathrm{E}\mathrm{n}\mathrm{d}}_{G_S}({\mathrm{i}\mathrm{n}\mathrm{d}}_{K_0}^{G_S}(\sigma ))$. We prove that it is free and of infinite rank.