The Derived Unipotent Block of $p$-Adic ${\mathrm{G}\mathrm{L}}_2$ as Perfect Complexes over a dg Schur Algebra

For a $p$-adic field $F$ of residual cardinality $q$, we provide a triangulated equivalence between the bounded derived category $D^b({\mathcal{B}}_1(G{)}_{fg})$ of finitely generated unipotent representations of $G={\mathrm{G}\mathrm{L}}_2(F)$ and perfect complexes over a dg enriched Schur algebra, in the non-banal case of odd characteristic $l$ dividing $q+1$. The dg Schur algebra is the dg endomorphism algebra of a projective resolution of a direct sum $V$ of the parahoric inductions of the trivial representations of the reductive quotients of $G$, and $V$ is shown to be a classical generator of $D^b({\mathcal{B}}_1(G{)}_{fg})$.