Bongartz–Ringel accessibility/no-gap conjecture

Let $A$ be an algebra and let $M$ be an indecomposable $A$-module of finite length. A module is accessible if simple modules are accessible and, inductively, an indecomposable module of length $n$ is accessible when it has an indecomposable submodule or an indecomposable quotient of length $n-1$.

Bongartz–Ringel accessibility conjecture. Every indecomposable module of finite length over any algebra is accessible.

This strengthens the result of Bongartz and Ringel that an algebra with an indecomposable module of length $n$ has an accessible module of length $n$; equivalently, the modules realizing the corresponding Brauer–Thrall-type statement should be obtainable by successive maximal submodules or quotients. The general assertion is presented here as a conjecture.