Affine Hecke algebra conjecture for metaplectic type theory

Let $\widetilde{G}$ be the metaplectic cover and $(\widetilde{J},\widetilde{\lambda})$ the type constructed in the paper. Let $\widetilde{M}$ be the associated Levi subgroup, $(\widetilde{J}_{M},\widetilde{\lambda}_{M})$ the corresponding type, $t_{P}$ the embedding of Hecke algebras, $\Psi_{0}(\mathcal{H}_{0}(t,\boldsymbol{q}_{0}))$ the finite part, and $\mathbb{C}[T_{0}]$ the lattice part identified with $t_{P}(\mathcal{H}(\widetilde{M},\widetilde{\lambda}_{M}))$. Affine Hecke algebra conjecture. The Hecke algebra is an affine Hecke algebra of type $A$, with an isomorphism of algebras

$$t_{P}(\mathcal{H}(\widetilde{M},\widetilde{\lambda}_{M}))\otimes_{\mathbb{C}}\Psi_{0}(\mathcal{H}_{0}(t,\boldsymbol{q}_{0}))\cong \mathcal{H}(\widetilde{G},\widetilde{\lambda}),$$

where the first factor is the lattice part and the second is the finite part, with the product structure on the left defined accordingly. This is presented as an expected general description of the Hecke algebra; the paper's preceding results establish the lattice component but do not establish the full asserted decomposition in general.