Geometric Howe correspondence conjecture for K(𝓧)

Let $(G,H)$ be a dual split reductive pair over $\mathbf{k}$, let $\mathcal{W}$ be the graded Weil category, and let $(D\mathcal{W})^{I_{G}\times I_{H}}$ denote its $I_{G}\times I_{H}$-invariant category. The Grothendieck group $K((D\mathcal{W})^{I_{G}\times I_{H}})$ is a bimodule for the affine extended Hecke algebras $\mathbb{H}_{G}$ and $\mathbb{H}_{H}$. Geometric Howe correspondence conjecture. The bimodule $K(\mathcal{X})$ is isomorphic to $K((D\mathcal{W})^{I_{G}\times I_{H}})$ under the action of $\mathbb{H}_{G}$ and $\mathbb{H}_{H}$. This is the paper's second conjecture, relating the geometric Howe correspondence to the K-theoretic construction; the source gives no resolution.