Howe correspondence conjecture for dual general linear pairs

Let $G=\mathbf{GL}_{n}$ and $H=\mathbf{GL}_{m}$ with $n\leq m$. Let $K(\mathcal{X})$ and $K(DP_{I_{H}\times I_{G}}(\Pi(F)))$ be the associated bimodules under the extended affine Hecke algebras $\mathbb{H}_{H}$ and $\mathbb{H}_{G}$. Howe correspondence conjecture. The bimodules $K(\mathcal{X})$ and $K(DP_{I_{H}\times I_{G}}(\Pi(F)))$ are isomorphic under the action of the extended affine Hecke algebras $\mathbb{H}_{H}$ and $\mathbb{H}_{G}$. This is the conjecture stated in the section on type-II dual reductive pairs; the source gives no resolution.