Does every separated Noetherian scheme have the resolution property?

A Noetherian scheme $X$ has the resolution property if every coherent $\mathcal O_X$-module $\mathcal F$ admits a surjection

$$\mathcal E\twoheadrightarrow\mathcal F$$

from a finite-rank vector bundle $\mathcal E$. This asks for a single vector-bundle quotient, not a finite locally free resolution.

Open problem. Does every separated Noetherian scheme have the resolution property? Equivalently, can one construct such an $X$ and a coherent sheaf $\mathcal F$ that is not a quotient of any finite-rank vector bundle? More sharply, does every proper scheme over a field have the resolution property?

Known boundary. The property holds for schemes with an ample family of line bundles, a class that includes quasi-projective schemes and regular separated Noetherian schemes. Gross proved it for every separated surface of finite type over a field. Mathur and Schröer proved that a reduced separated excellent scheme has the property away from a closed subset of codimension at least three.

Thus a finite-type counterexample over a field would have to be singular, non-quasi-projective, and at least three-dimensional; a proper counterexample would also have to be nonprojective. The proper case is still explicitly listed as open by the Stacks Project.