Acyclic Ext-quiver conjectures for fractionally Calabi–Yau and representation-finite algebras

Let $A$ be an algebra of finite global dimension. Its Ext-quiver is the quiver encoding the relevant extension data between the simple $A$-modules; it is acyclic if it has no oriented cycles.

Acyclic Ext-quiver conjectures.

1. If $A$ is twisted fractionally Calabi–Yau, then $A$ has an acyclic Ext-quiver.
2. If $A$ is $d$-representation-finite, then $A$ has an acyclic Ext-quiver.

These are folklore open problems concerning structural properties of twisted fractionally Calabi–Yau and $d$-representation-finite algebras. The source formulates both assertions as conjectures, and no resolution is given.