The partial-converse conjecture for primitive ideals of local functions

Let $\g$ be one of $\W{-1}$, $W$, or $\Vir$, and let $\chi$ and $\eta$ be one-point local functions on $\g$. Let $P(\chi)$ and $P(\eta)$ denote their associated Poisson primitive ideals, and let $Q_\chi$ and $Q_\eta$ denote the corresponding primitive ideals of $\Ua(\g)$. Partial-converse conjecture. One has

$$P(\chi)\subsetneqq P(\eta)\quad\Longleftrightarrow\quad Q_\chi\subsetneqq Q_\eta.$$

This conjecture proposes that strict inclusion among the associated Poisson primitive ideals is exactly reflected by strict inclusion among the corresponding enveloping-algebra primitive ideals; the surrounding text notes that the converse of the previously established inclusion proposition is false in general, motivating this restricted one-point version.