Projectivity conjecture for noncommutative Hilbert schemes

Let $R$ be a connected graded, strongly noetherian ring satisfying strong $\chi$, and let $P$ be a finitely graded, noetherian $R$-module. The quotients of $\pi P$ in $\operatorname{qgr} R$ with fixed Hilbert function $h$ are parameterized by a countable union of commutative projective schemes.

Projectivity conjecture. This union of schemes is actually a projective scheme.

The preceding theorem gives a countable union of commutative projective schemes parameterizing these quotients; the conjecture asserts that the entire parameter space is itself projective. Its resolution is not indicated in the supplied text.