Regularity conjecture for antiunitary representations of simply connected Lie groups

Let $G$ be a simply connected Lie group, let $\tau_h$ be the involution associated with an Euler element $h\in\mathfrak g$, and let $(U,\mathcal H)$ be an antiunitary representation of $G_{\tau_h}$. Write ${\tt V}(h,U)$ for the associated standard subspace. The representation is $h$-regular when there exists an $e$-neighborhood $N\subeq G$ such that

$${\tt V}_N:=\bigcap_{g\in N}U(g){\tt V}(h,U)$$

is cyclic. Regularity conjecture. If $G$ is simply connected and $h\in\mathfrak g$ is an Euler element, then every antiunitary representation of $G_{\tau_h}$ is $h$-regular. This conjecture asks for universal regularity of the standard subspaces arising from antiunitary representations; the paper presents it as an open problem, while earlier results establish regularity in particular settings.