The supercritical NLS standing-ring blowup conjecture

Let $\psi$ be a singular standing-ring solution of the NLS equation with $d>1$ and $\sigma>2$. Let $L(t)$ be its ring-region similarity scale, $r_{\max}(t)$ its ring radius, and $\psi_F$ the self-similar profile. Supercritical NLS standing-ring conjecture. The solution is self-similar in the ring region, with $|\psi|\sim|\psi_F|$ for $r-r_{\max}=\mathcal{O}(L)$; $\psi_F(t,r)=\phi_S(t,x=r-r_{\max}(t))$, where $\phi_S$ is the asymptotic peak-type profile of the one-dimensional NLS with exponent $\sigma$, and the admissible profile has $\kappa=\kappa_S(\sigma)$ and $S_0=S_0(\sigma)$. Moreover,

$$L(t)\simkappa_S(\sigma)\sqrt{T_c-t},\qquad t\longrightarrow T_c.$$

The claim identifies higher-dimensional standing-ring collapse with one-dimensional supercritical peak collapse; the parser supplies no resolution evidence, so it remains open.