Rigidity conjecture for cluster tilting modules over local commutative algebras

Let $A$ be a local commutative algebra with an $n$-cluster tilting module. Rigidity conjecture. Then $n=1$ and there is an integer $m\geq 1$ such that

$$A\cong K[x]/(x^m).$$

The conjecture is one of the paper's experimentally motivated claims about local algebras; the supplied text gives no resolution, so it remains open.