The strong Dixmier-map surjectivity conjecture for Witt and Virasoro algebras

Let $\g$ be one of $\W{-1}$, $W$, or $\Vir$. Let $\operatorname{P.Prim}\operatorname{S}(\g)$ denote the Poisson primitive spectrum of the symmetric algebra and let $\operatorname{Prim}\operatorname{U}(\g)$ denote the primitive spectrum of the enveloping algebra. Strong Dixmier-map surjectivity conjecture. The strong Dixmier map

$$\widetilde{\operatorname{Dx}}^\g:\operatorname{P.Prim}\operatorname{S}(\g)\longrightarrow\operatorname{Prim}\operatorname{U}(\g)$$

is surjective. If true, this would yield further structural consequences for primitive ideals, including complete primeness, the ascending chain condition, and descriptions by kernels of maps to Weyl-type algebras.