The 3-adic Steinberg block and a categorical 3-adic local Langlands correspondence for ${\text{GL}}_2({\mathbb{Q}}_3)$

We extend the work of Johansson-Newton-Wang-Erickson on a categorical $p$-adic Langlands correspondence over ${\text{GL}}_2({\mathbb{Q}}_p)$ to $p=3$ by recontextualizing their construction in the Steinberg case. We produce a novel, more categorical method of analyzing the properties of this construction without requiring an explicit algebraic presentation. As input to this argument, we also perform some new analysis of the representation theory of the $3$-adic Steinberg block of ${\text{GL}}_2({\mathbb{Q}}_3)$, obtaining results complementing those of Paškūnas.