Equicontinuity conjecture for flows conserving the perturbation determinant

Let $\mathcal{S}$ denote the Schwartz class, and let $a(\kappa;q)$ be the perturbation determinant defined for $q\in L^2$ and $\kappa\geq1$. Suppose that

$$\Phi:\mathbb{R}\times\mathcal{S}\to\mathcal{S}$$

is a well-posed flow conserving $a(\kappa;q)$ for every $\kappa\geq1$. For a set $Q\subseteq\mathcal{S}$, call $Q$ $L^2$-equicontinuous when its elements have uniformly small high-frequency tails, and define

$$Q_* = \{ \Phi(t,q): q\in Q \text{ and } t\in\mathbb{R}\}.$$

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

This is the grand version of the preceding conjecture and is intended to apply to a wide range of commuting flows. The stated theorem proves the analogous conclusion under the mass restriction $\sup_{q\in Q}\|q\|_{L^2}^2<4\pi$, while the unrestricted assertion remains open.