Independence conjecture for Auslander regular incidence algebras

Let $K$ be a field of characteristic $0$, and let $P$ be a poset. Denote its incidence algebra by $KP$, and let independent and the Coxeter permutation be the corresponding notions for $P$. Independence conjecture for Auslander regular incidence algebras. If the incidence algebra $KP$ is Auslander regular, then $P$ is independent and the Coxeter permutation coincides with the Auslander–Reiten permutation. This conjecture would show that posets with Auslander regular incidence algebras are independent. It was posed as an open conjecture in earlier work; the supplied text gives no resolution.