The $L^2$-supercritical stochastic NLS blow-up conjecture

Let $u_0\in H^1(\mathbb R)$ and let $u(t)$ be an evolution of the stochastic nonlinear Schrödinger equation with $\sigma>2$ and the specified space-time white-noise coefficient. Let $Q=Q_{1,0}$ be the blow-up profile solving the referenced profile equation, and let $a$ be the corresponding constant. The $L^2$-supercritical blow-up conjecture. In the multiplicative (Stratonovich) noise case, sufficiently localized initial data blows up in finite positive random time with positive probability. In the additive noise case, any initial data leads to blow-up in finite random time almost surely. If a solution blows up at a random positive time $T(\omega)>0$, then, for $t$ close to $T(\omega)$,

$$u_c(t,x)\sim L(t)^{-1/\sigma}Q\left(\frac{x-x(t)}{L(t)}\right)\exp\left(i\theta(t)+\frac{i}{2a(t)}\log\frac{T}{T-t}\right),$$

where $a(t)\to a$, $\theta(t)\to\theta_0$, $x(t)\to x_c$, and $L(t)=(2a(T-t))^{1/2}$. Consequently,

$$\\|\nabla u(t,\cdot)\\|_{L_x^2}\sim L(t)^{-(1-s)}\equiv (2a(T-t))^{-\frac12(\frac12+\frac1\sigma)}.$$

Conditionally on finite-time blow-up, $x_c$ is Gaussian; no conditioning is necessary in the additive case. Thus the blow-up has a polynomial rate without correction and the deterministic supercritical blow-up profile. The source presents this as a conjecture and reports numerical confirmation, but does not supply a general rigorous resolution.