Przebinda's character correspondence conjecture via the Chc-star transform

Let $({\rm G},{\rm G}')$ be an irreducible dual pair in ${\rm Sp}(W)$ with ${\rm rk}({\rm G})\leq {\rm rk}({\rm G}')$. Let ${\rm Chc}^{*}$ be the map transferring invariant distributions from $\widetilde{{\rm G}}$ to $\widetilde{{\rm G}'}$, and let ${\rm G}_{1}$ and ${\rm G}'_{1}$ be the Zariski identity components of ${\rm G}$ and ${\rm G}'$. For $\Pi\in\mathscr{R}(\widetilde{{\rm G}},\omega)$, assume that ${\Theta_{\Pi}}_{|_{\widetilde{{\rm G}}/\widetilde{{\rm G}_{1}}}}=0$ if ${\rm G}={\rm O}({\rm V})$, where ${\rm V}$ is an even-dimensional vector space over $\mathbb{R}$ or $\mathbb{C}$. Przebinda's character correspondence conjecture. Up to a constant, the character corresponding to $\Pi$ under theta correspondence satisfies

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

onetheless on $\widetilde{{\rm G}'_{1}}$. The conjecture predicts that character correspondence in the theta correspondence is obtained through the ${\rm Chc}^{*}$ transform; the statement is attributed to T. Przebinda, and no resolution is supplied in the source.