Equicontinuity conjecture for derivative nonlinear Schrödinger flows

Let $\mathcal{S}$ denote the Schwartz class. For a set $Q\subseteq\mathcal{S}$, call $Q$ $L^2$-equicontinuous if its elements have uniformly small high-frequency tails, and let $e^{tJ\nabla H}q$ denote the derivative nonlinear Schrödinger flow from initial data $q$. Define

$$Q_* = \{ e^{tJ\nabla H} q: q\in Q \text{ and } t\in\mathbb{R}\}.$$

Equicontinuity conjecture. For any $Q\subseteq\mathcal{S}$ that is $L^2$-bounded and equicontinuous, the set $Q_*$ is also $L^2$-equicontinuous.

This conjecture would rule out the concentration scenario associated with type-II blowup and is motivated by the fact that the mass is a coercive, scaling-critical conserved quantity. Local well-posedness is known for $H^s$ data with $s\geq\frac12$, but global well-posedness at the $L^2$ level is not known.