Tau-rigidity conjecture for sums of an indecomposable syzygy and its involutive translate

Let $A$ be a dimer tree algebra with an automorphism $c3$ of order $2$, and let $M$ be an indecomposable syzygy over $A$. The direct sum $M\oplusc3M$ is formed using the action of this involution.

Tau-rigidity conjecture. If $M$ is an indecomposable syzygy, then $M\oplusc3M$ is $c4$-rigid.

The geometric model already shows that $M\oplusc3M$ is rigid. The conjecture asks for the stronger $c4$-rigidity property, equivalently related in the paper to $c4$-rigidity after induction to the skew group algebra.