Pro-$p$ Iwahori-Hecke modules in semisimple rank one and singularity categories

Let $\mathfrak{F}$ be a non-archimedean local field of residue characteristic $p$ and $G$ be one of the groups $\mathrm{GL}_2(\mathfrak{F})$, $\mathrm{SL}_2(\mathfrak{F})$ or $\mathrm{PGL}_2(\mathfrak{F})$. Let $\mathcal{H}_G$ denote the pro-$p$ Iwahori-Hecke algebra of $G$ over $\overline{\mathbb{F}}_p$. We study the homotopy category $\mathrm{Ho}(\mathcal{H}_G)$ of Hovey's Gorenstein projective model structure on the category of $\mathcal{H}_G$-modules and relate it to the singularity category $\mathrm{Sing}(X_{q,\mathbf{G}})$ of an explicit scheme. When $G=\mathrm{GL}_2(\mathfrak{F})$, this scheme was first introduced by Dotto-Emerton-Gee \cite{DEG22}. We obtain in that case an equivalence $\mathrm{Ho}(\mathcal{H}_{\mathrm{GL}_2})\simeq \mathrm{Sing}(X_{q,\mathrm{GL}_2})$ and recover from this Grosse-Kl\"onne's mod-$p$ Langlands correspondence for Hecke modules \cite{GK20}, building on work of P\'epin-Schmidt \cite{PeSch25_2}. We furthermore describe $\mathrm{Ho}(\mathcal{H}_G)$ completely explicitly when $G=\mathrm{SL}_2(\mathfrak{F})$ or $\mathrm{PGL}_2(\mathfrak{F})$, and make additional computations in the $\mathrm{GL}_2$ case.