Koszulness conjecture for De Concini–Procesi wonderful models

A De Concini–Procesi wonderful model is the smooth projective model associated with a building set of a subspace arrangement. Consider a subspace arrangement in $\mathbb{P}(V)$ that refines a hyperplane arrangement. Let $\mathcal{G}$ be a building set of the corresponding lattice of subspaces, and let $\overline{Y}_{\mathcal{G}}$ be the associated De Concini–Procesi projective wonderful model.

Wonderful-model Koszulness conjecture. The ring

$$H^\bullet(\overline{Y}_{\mathcal{G}},\mathbb{Z})$$

is Koszul if and only if it is quadratic.

This conjecture proposes that, for this broad class of wonderful models, failure of quadraticity is the only obstruction to Koszulness. The paper also conjectures Koszulness for De Concini–Procesi models in a wide range of cases, while the general assertion remains open.