Shrinking-ring blowup conjecture for the biharmonic nonlinear Schrödinger equation

Let $d>1$ and $4/d<\sigma<4$, and let $\psi$ be a singular ring-type solution of the biharmonic nonlinear Schrödinger equation (BNLS). Let $T_{\rm c}$ be the collapse time, $r_{\rm \max}(t)$ the ring radius, and $L(t)$ the ring width. Define

$$\alpha_{\mathrm B}=\frac{4-\sigma}{\sigma(d-1)}.$$

Shrinking-ring conjecture. The solution is quasi self-similar, $\psi\sim\psi_{Q_{\mathrm B}}$, for $r-r_{\rm \max}=\mathcal{O}(L)$; it is a shrinking ring, with $\lim_{t\to T_{\rm c}}r_{\rm \max}(t)=0$; its shrinking rate is $\alpha=\alpha_{\mathrm B}$, where $0<\alpha<1$; and

$$L(t)\sim\kappa(T_{\rm c}-t)^p,\qquad p=\frac{1}{3+\alpha_{\mathrm B}}=\frac{1}{4-\frac{\sigma d-4}{\sigma(d-1)}},$$

where $1/4<p<1/3$.

The conjecture predicts strong, shrinking-ring collapse in the supercritical range $4/d<\sigma<4$ and a blowup rate interpolating between the quartic-root regime and the rate at the upper endpoint. The source presents this as an analogy-based prediction supported by numerical experiments, with no proof or resolution stated.