The weak maximal self-orthogonal conjecture for projectively Wakamatsu tilting modules

Let $\Lambda$ be an artin algebra. A module is projectively Wakamatsu tilting if it satisfies the projective Wakamatsu tilting conditions, and maximal self-orthogonal if it cannot be enlarged by a non-isomorphic indecomposable direct summand while remaining self-orthogonal.

Weak maximal self-orthogonal conjecture. Every projectively Wakamatsu tilting module is maximal self-orthogonal.

This is a weaker variant of the maximal self-orthogonal conjecture, replacing Wakamatsu tilting by projectively Wakamatsu tilting. The paper relates it to other homological conjectures, including the Auslander–Reiten conjecture.