The maximal-rigid-to-cluster-tilting conjecture

Let $\operatorname{\mathcal C}$ be a connected Hom-finite triangulated 2-Calabi–Yau category. A maximal rigid object is a rigid object maximal under adding direct summands, and its quiver is the associated endomorphism quiver.

The maximal-rigid-to-cluster-tilting conjecture. Any maximal rigid object whose quiver has no loops or 2-cycles is a cluster tilting object.

Maximal rigid objects need not be cluster tilting in general, so the absence of loops and 2-cycles is proposed as a sufficient condition. The statement is presented as a suspicion and remains unproved in the supplied text.