Perelman's codimension-one manifold conjecture for non-generic NLS blow-up

Consider the critical focusing nonlinear Schrödinger equation in one spatial dimension and its standing-wave blow-up solutions. The generalized root space of the linearized operator has five modes associated with internal symmetries and one additional exotic mode that cannot be controlled by those symmetries. Perelman's conjecture. There exists a co-dimension $1$ manifold of initial data resulting in the non-generic blow-up behavior. A modulation-theoretic treatment of the internal symmetries should control the modes associated with those symmetries, leaving one unstable direction and hence a codimension-one stable manifold. The statement is presented as the expected improvement of the Bourgain–Wang construction.