Improved uniqueness of multi-breathers of the modified Korteweg-de Vries equation

We consider multi-breathers of (mKdV). Previously, a smooth multi-breather was constructed, and proved to be unique in two cases: first, if the class of super-polynomial convergence to the profile, and second, under the assumption that all speeds of the breathers involved are positive (without rate of convergence). The goal of this short note is to improve the second result: we show that uniqueness still holds if at most one velocity is negative or zero.