Codimension-one manifold conjecture for near-soliton wave maps

Consider equivariant wave-map solutions that are close to a soliton in a stronger topology, with additional regularity and decay at infinity. Type 1 solutions are finite-time blow-up solutions, Type 3 solutions satisfy $\lambda(t)\to 0$ as $t\to\infty$, and Type 4 solutions have a scale parameter $\lambda(t)$ that remains in a compact set for all time. Codimension-one manifold conjecture. There exists a codimension-one set of small data leading to Type 4 solutions, which separates Type 1 and Type 3 solutions. This would describe a separating threshold among near-soliton solutions, complementing the known constructions of Type 1 solutions; the existence of such a set was posed as a starting point for understanding dynamics near a soliton.