Non-generic components of the Emerton-Gee stack for ${\mathrm{G}\mathrm{L}}_2$

Let $K$ be a finite unramified extension of ${\mathbb{Q}}_p$ with $p>3$. We study the extremely non–generic irreducible components in the reduced part of the Emerton–Gee stack for ${\mathrm{G}\mathrm{L}}_2$. We show precisely which irreducible components are smooth, which are normal, and which have Gorenstein normalizations. We show that the normalizations of the irreducible components admit smooth–local covers by resolution–rational schemes. We also determine the singular loci on the components, and use our results to update expectations about the conjectural categorical $p$–adic Langlands correspondence.