Primitive and prime spectrum equivalence for the Witt algebra and its positive part

Let $W$ be the Witt algebra and $W_{\geq -1}$ its subalgebra spanned by the basis elements of index at least $-1$. Let $\operatorname{U}(W)$ and $\operatorname{U}(W_{\geq -1})$ be their universal enveloping algebras, and let $\operatorname{PSpec}_{\rm prim}$ denote the primitive spectrum. The Witt-spectrum restriction conjecture. Restriction gives a bijection between primitive, respectively prime, ideals of $\operatorname{U}(W)$ and $\operatorname{U}(W_{\geq -1})$, and induces a homeomorphism

$$\operatorname{PSpec}_{\rm prim}\operatorname{U}(W)\stackrel{\sim}{\longrightarrow}\operatorname{PSpec}_{\rm prim}\operatorname{U}(W_{\geq -1}).$$

The conjecture is suggested by corresponding results for Poisson spectra, but its resolution is not stated in the supplied text.