Existence of dispersion-manageable managed NLS data

Let $(NLS)_\gamma$ and $(NLS)^\gamma$ denote the two managed nonlinear Schrödinger models, and call a solution dispersion-manageable when it exists globally and there are constants $c_0,C_0\in(0,\infty)$ such that

$$c_0\leq \sup_{t\geq 0}\left\|u(t,\cdot)\right\|_{L_x^\infty({\mathbb R}^d)}\leq C_0.$$

Dispersion-manageability conjecture. There exists a Schwartz function $u_0$ such that, with $u(0,x)=u_0(x)$, either $(NLS)_\gamma$ or $(NLS)^\gamma$ is dispersion-manageable.

The property describes global, non-scattering behavior and is presented as a numerically motivated stabilization phenomenon. The supplied text does not state whether the conjecture has been resolved.