The distributivity conjecture for lattices with pure minimal injective coresolutions

Let $L$ be a finite lattice, and let $A=KL$ be its incidence algebra. Assume that $A$ has a pure minimal injective coresolution. Distributivity conjecture. Then $L$ is distributive. The conjecture is motivated by the relationship between pure minimal injective coresolutions and Auslander regularity, and it has been verified for all finite lattices with at most 10 elements; its general case remains open.