Improved Uniqueness of Multi-breathers of the Modified Korteweg-de Vries Equation

We consider multi-breathers of (mKdV). In Semenov (2022), a smooth multi-breather was constructed, and proved to be unique in two cases: first, in the class of super-polynomial convergence to the profile (in the spirit of (Commun. Partial Differ. Equ. 46, 2325-2385, 2021)), 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.