Surjectivity of Kirillov's orbit method for Virasoro and Witt algebras

Let $\mathfrak g$ be one of $V\!ir$, $W$, or $W_{\geq -1}$. For a local function $\chi\in\mathfrak g^*$, choose a polarization and form the annihilator of the induced module; write $\mathbb O(\chi)$ for its pseudo-orbit. Let $\operatorname{PSpec}_{\rm prim}\operatorname{S}(\mathfrak g)$ and $\operatorname{Spec}_{\rm prim}\operatorname{U}(\mathfrak g)$ denote the corresponding Poisson-primitive and primitive spectra. The orbit-method surjectivity conjecture. Kirillov's assignment from $\chi$ to a polarization and then to the annihilator of the induced module always produces a primitive ideal, is independent of the polarization, depends only on $\mathbb O(\chi)$, and induces a surjective map

$$\operatorname{PSpec}_{\rm prim}\operatorname{S}(\mathfrak g)\longrightarrow\operatorname{Spec}_{\rm prim}\operatorname{U}(\mathfrak g).$$

This would establish an orbit-method description of all primitive ideals for these infinite-dimensional Lie algebras. The paper verifies the construction for particular two-dimensional pseudo-orbits, while the full assertion remains open.