Three-quasi-boxes rigidity conjecture for fully reduced rank 2 modules

Let $M\in {\rm CM}(B_{k,2k})$ be a fully reduced rank $2$ module. Suppose that the rims of $M$ form at least three quasi-boxes, but not exactly three boxes.

Three-quasi-boxes rigidity conjecture. Then $M$ is not rigid.

The claim is part of the proposed characterization of rigid indecomposable rank $2$ modules: exactly three boxes imply rigidity, whereas the remaining fully reduced configurations with at least three quasi-boxes are expected not to be rigid. The supplied text does not establish the assertion in full.