Koszulity conjecture for Chow rings of flag or complete built matroids

Let $\mathrm{M}$ be a matroid and let $\mathcal{G}$ be a building set such that $(\mathrm{M},\mathcal{G})$ is a flag or complete built matroid. Koszulity conjecture. The Chow ring of $(\mathrm{M},\mathcal{G})$ is Koszul.

This generalizes known Koszulity results for maximal building sets, augmented Chow rings, supersolvable built lattices, and the braid-matroid case with minimal building set. The source presents the general statement as open.