Bernon–Przebinda character-transfer conjecture

Assume that $\operatorname{rk}({\rm G})\leq\operatorname{rk}({\rm G}')$. Let ${\rm G}_{1}$ and ${\rm G}'_{1}$ be the Zariski identity components of ${\rm G}$ and ${\rm G}'$, respectively, and let

$${\rm Chc}^{*}:\mathscr{D}'(\widetilde{{\rm G}})^{{\widetilde{{\rm G}}}}\to\mathscr{D}(\widetilde{{\rm G}'})^{{\widetilde{{\rm G}'}}}$$

be the Cauchy-Harish-Chandra transfer map. Let $\Pi\in\mathscr{R}(\widetilde{{\rm G}},\omega)$ satisfy

$${\Theta_{\Pi}}_{|_{\widetilde{{\rm G}}/\widetilde{{\rm G}_{1}}}}=0$$

if ${\rm G}={\rm O}({\rm V})$, where ${\rm V}$ is an even-dimensional vector space over $\mathbb{R}$ or $\mathbb{C}$.

Bernon–Przebinda's conjecture. Up to a constant,

$${\rm Chc}^{*}(\Theta_{\Pi})=\Theta_{\Pi'_{1}}$$

on $\widetilde{{\rm G}'_{1}}$.

The claim gives a character-level description of Howe correspondence via the Cauchy-Harish-Chandra transfer. The supplied text does not state whether this formulation has been resolved, so its database status remains open.