The stable Morita-equivalence conjecture for rational representations of $GL_2$

Let $F$ be the field in which rational representations of $GL_2(F)$ are considered, and let $\underleftarrow{\mathcal{C}}_p(F)\operatorname{-mod}$ denote the corresponding module category. A block is an indecomposable direct summand of the rational representation category. The stable $GL_2$ conjecture. Every block of rational representations of $GL_2(F)$ is equivalent to $\underleftarrow{\mathcal{C}}_p(F)\operatorname{-mod}$. This prediction is presented as a strengthening of the preceding Morita-equivalence theorem for the associated graded algebra; the source gives no resolution.