Central-character conjecture for dual conjugacy classes

Let $G_{\mathbb C}$ be a complex group with real form $G_{\mathbb R}$, and let $\mathfrak C_{\mathbb C}$ be a unipotent conjugacy class, or a semisimple conjugacy class obtained by deforming such a class. Let $\mathfrak C_{\mathbb R}\subset\mathfrak C_{\mathbb C}$ be the corresponding real orbit, and let $\dual{\mathfrak C}_{\mathbb C}$ be the dual conjugacy class with semisimple element $\dual S$. Central-character conjecture. The parameter

$$\frac{1}{2\pi}\log \dual S$$

of the dual conjugacy class $\dual{\mathfrak C}_{\mathbb C}$ is equal to the central character of any $G_{\mathbb R}$-representation obtained by quantizing $\mathfrak C_{\mathbb R}$. This conjecture predicts that the duality map on conjugacy classes records the central character produced by orbit quantization, extending the comparison between dual conjugacy-class data and representation-theoretic invariants.