Perverse character sheaf conjecture for loop Lie algebras

Let $G$ be a reductive group, let $\xi=(M,\lambda,L,\mathcal{O},\mathcal{L})$ be a polar-cuspidal datum for $G$, and let $\mathbf{Q}\subset LM$ be a parahoric subgroup containing $L$ as a Levi factor. Let $L\mathfrak{g}^{\heartsuit}\subset L\mathfrak{g}$ denote the regular semisimple locus, and let $\mathcal{S}_{\xi,\mathbf{Q}}$ be the associated character sheaf. Perverse character sheaf conjecture. The restriction

$$\mathcal{S}_{\xi,\mathbf{Q}}|_{L\mathfrak{g}^{\heartsuit}}\in D^\bullet_{LG}(L\mathfrak{g}^{\heartsuit})$$

is perverse in the sense of Bouthier--Kazhdan--Varshavsky. The conjecture extends the known perversity result for the affine Springer sheaf on the regular semisimple locus, while its validity for general polar-cuspidal data and parahoric subgroups remains open.