Echelonmotion and the Auslander–Reiten permutation for Auslander-regular posets

Let $R$ be a finite poset whose incidence algebra over a field of characteristic $0$ is Auslander-regular, and suppose that $R$ is connected. A finite poset is echelon-independent when the echelon operator is independent of the choice of linear extension, and echelonmotion is the resulting operator on $R$. The Auslander–regular poset conjecture. $R$ is echelon-independent, and echelonmotion on $R$ is the inverse of the Auslander–Reiten permutation of $R$. The classification of Auslander-regular posets is open, with partial progress known; this conjecture would connect echelonmotion with homological algebra beyond the distributive-lattice case.