Uniform Lipschitz regularity of flat segregated interfaces in a singularly perturbed problem

For the singularly perturbed system $$\Delta u_{i,\beta}=\beta u_{i,\beta}\sum_{j\neq i}u_{j,\beta}^2, \quad 1\leq i\leq N,$$ we prove that flat interfaces are uniformly Lipschitz. As a byproduct of the proof we also obtain the optimal lower bound near the flat interfaces, $$\sum_iu_{i,\beta}\geq c\beta^{-1/4}.$$