Uniform smoothness conjecture for soliton addition and removal maps

Let $B^N_+:H^s\times\mathbf M_N\to H^s$ be the soliton addition map and $B^N_-:H^s\to H^s\times\mathbf M_N$ the soliton removal map for the cubic nonlinear Schrödinger equation. Here $\mathbf M_N$ denotes the $N$-soliton set, and $H^s$ is the Sobolev space in which these maps are defined.

Uniform smoothness conjecture. The maps $B^N_+$ and $B^N_-$ are uniformly smooth on bounded sets in $H^s$.

Uniform smoothness would complete the global regularity theory for the soliton addition and removal parametrization. The local smoothness and inverse-map properties are established in the paper, but uniform smoothness in full generality remains open.