Ind-coherent Hecke equivalence conjecture for the $\GL(n)$ theory

Let

$$\calY=\left(\prod_{i=1}^{n-1}\Hom(\mathbb C^i,\mathbb C^{i+1})\right)\bigg/\prod_{i=1}^{n-1}\GL(i)$$

be the stack specified in the source. Let $\IndCoh$ denote ind-coherent sheaves and let $D(\Gr_{\GL(n)})^{\operatorname{Hecke}}$ be the Hecke-eigenmodule category on the affine Grassmannian. Ind-coherent Hecke equivalence conjecture. The category $\IndCoh(\Maps(\calD^*_{dR},\calY))$ is equivalent to $D(\Gr_{\GL(n)})^{\operatorname{Hecke}}$. This is a corrected form of the preceding naive $\QCoh$ expectation, motivated by the distinction between quasi-coherent and ind-coherent sheaves; the source gives no resolution.