Auslander–Reiten map characterisation of Auslander-Gorenstein algebras

Let $A$ be a finite-dimensional $K$-algebra over an algebraically closed field $K$. Say that $A$ has a well-defined Auslander–Reiten map if every indecomposable injective $A$-module has a finite minimal projective resolution whose last term is an indecomposable projective module. Auslander–Reiten map conjecture. The algebra $A$ is Auslander-Gorenstein if and only if it has a well-defined Auslander–Reiten map that is a bijection. This conjecture proposes a converse to the Auslander–Reiten bijection for Auslander-Gorenstein algebras and would characterise such algebras through their injective resolutions.