The inclusion conjecture for primitive ideals of local functions

Let $\g$ be one of $\W{-1}$, $W$, or $\Vir$, and let $\chi$ and $\eta$ be local functions on $\g$. Let $P(\chi)$ and $P(\eta)$ denote the associated Poisson primitive ideals, and let $Q_\chi$ and $Q_\eta$ denote the corresponding primitive ideals of $\Ua(\g)$. Inclusion conjecture. Strict inclusion of the Poisson primitive ideals should imply strict inclusion of the corresponding primitive ideals:

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

This extends the one-point statement to multi-point local functions. The source explicitly expects the converse implication to be false.