Symplectic singularities of Lafforgue’s functoriality variety

Determine whether the affine closure of the Lafforgue functoriality variety Xi_{H→G}, defined in equation (vin0) as Xi_{H→G} = J_{\check H} \times^{\varrho^* J_{\check H \times \check G}}_{\_} T^*_{\psi}(\check G/\check U) (equivalently, the twist of T^*_{\psi}(\check G/\check U) by the torsor over \check{\mathfrak g}^* described there), has symplectic singularities; i.e., prove or refute that its affine closure \overline{\Xi_{H→G}} is a variety with symplectic singularities.

Background

Within Section 2, the paper constructs a quasi-affine smooth symplectic variety Xi_{H→G} associated to a morphism of connected reductive groups H→G, following Lafforgue’s functoriality. Xi_{H→G} is equipped with Hamiltonian actions and admits a moment map to the dual Lie algebra. The authors then prove that the affine closure \overline{\Xi_{H→G}} is a finitely generated, integrally closed domain and identify it with a spectrum of equivariant cohomology in Theorem \ref{vincent}.

They note several cases where symplectic singularities are known: for Borel subgroups (via Gannon’s result), for arbitrary parabolics in type A (through Coulomb branches), and more generally for all Coulomb branches (Bellamy’s theorem). They then pose the question of whether the same symplectic singularity property holds for \overline{\Xi_{H→G}} in full generality.