Nonlinear instability conjecture for traveling waves with decreasing myosin mass

Consider the traveling wave solutions of the free-boundary cell-motility model, parametrized by their velocity $V$, and let $M$ denote the conserved total myosin mass

$$M=\tilde{\Lambda}\int_\Omega e^{\Phi-Vx}\,dx\,dy.$$

Nonlinear instability conjecture. In the range of parameters such that the total myosin mass $M$ of traveling wave solutions decreases as a function of $V$ for small $V$, these solutions are nonlinearly unstable.

This conjecture concerns the stability change associated with the bifurcation from steady states to traveling waves. The source states that it is suggested by numerical results and will be rigorously proved in forthcoming work; no resolution is supplied here.