First-level modules are preserved by Auslander–Reiten translation

Let $M$ be an indecomposable module of the first level, and let $\tau M$ denote its Auslander–Reiten translation.

First-level Auslander–Reiten conjecture. The module $\tau M$ is also of the first level.

The authors state that an internal characterization of modules of the first level is unavailable, but regard this preservation property as very plausible. No resolution is given in the source.