The converse Wakamatsu tilting conjecture for maximal-size self-orthogonal modules

Let $\Lambda$ be an artin algebra, and let $T$ be a self-orthogonal $\Lambda$-module. Write $|T|$ for the number of non-isomorphic indecomposable direct summands of a basic module $T$, and $|\Lambda|$ for the number of isomorphism classes of simple $\Lambda$-modules.

Converse Wakamatsu tilting conjecture. If $|T|=|\Lambda|$, then $T$ is Wakamatsu tilting.

The conjecture proposes that a self-orthogonal module attaining the maximal expected size must be Wakamatsu tilting. It complements the boundedness and Proj=Inj conjectures and is presented as an additional open question.