Conjecture on the orbit stratification of the convolution Grassmannian

Let $G$ be the group appearing in the Pappas–Rapoport model, let $L^+G$ be its positive loop group, and let $\widetilde{\operatorname{Gr}}_{\mu_{\bullet}}$ be the convolution Grassmannian associated with the tuple of coweights $\mu_{\bullet}$. Write

$$\operatorname{Hecke}_{\mu_{\bullet}}=\big[L^+G\backslash \widetilde{\operatorname{Gr}}_{\mu_{\bullet}}\big]$$

and let $|\operatorname{Hecke}_{\mu_{\bullet}}|$ denote its underlying topological space.

Orbit-stratification conjecture. There exists a finite partially ordered set $(X,<)$ describing the $L^+G$-orbits in $\widetilde{\operatorname{Gr}}_{\mu_{\bullet}}$. Equivalently, there is a homeomorphism

$$|\operatorname{Hecke}_{\mu_{\bullet}}|\simeq X,$$

where $X$ is endowed with the topology induced by $<$.

This conjecture asserts that the orbit decomposition of the convolution Grassmannian gives a finite stratification whose underlying topological space is the finite poset of orbit strata. The supplied text does not state whether the conjecture has been proved or disproved.