The regularity conjecture for antiunitary representations

Let $G$ be a Lie group with Lie algebra ${\mathfrak g}$, let $h\in{\mathfrak g}$ be an Euler element, and let $G_{\tau_h}$ denote the subgroup associated with the involution determined by $h$. An antiunitary representation is a representation $(U,\mathcal{H})$ of $G_{\tau_h}$ by unitary and antiunitary operators on a Hilbert space $\mathcal{H}$. Regularity conjecture. Any antiunitary representation $(U,\mathcal{H})$ of $G_{\tau_h}$ is $h$-regular, meaning that there exists an open $e$-neighborhood $N\subseteq G$ such that

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

is cyclic. This conjecture asks whether the regularity required in the Euler Element Theorem holds for all antiunitary representations of $G_{\tau_h}$ without additional structural assumptions on $G$. It is described in the source as a challenging open problem; the cited cases include the $4$-dimensional split oscillator group.