Conjecture on the effect of quadratic phase on global existence and blow-up

Let $u_0$ belong to the space $\mathcal X$ or to $H^1$, and for $b\in\mathbb R$ define

$$v_0=e^{\frac{i b |x|^2}{4}}u_0.$$

Quadratic-phase conjecture. For the combined nonlinear Schrödinger equation, if $b>0$, the solution with initial condition $v_0$ exists globally and scatters, whereas if $b<0$ and $u_0$ is sufficiently localized, the solution with initial condition $v_0$ blows up in finite time.

This conjecture concerns the contrasting effects of the quadratic phase in the regimes $b>0$ and $b<0$. The preceding results establish global existence and scattering when $b>0$ is sufficiently large, while the paper reports numerical confirmation for both signs of $b$; the full assertion is not established in the supplied context.