Equal-rate blowup conjecture for critical biharmonic nonlinear Schrödinger rings

Let $\psi$ be a ring-type singular solution of the critical biharmonic nonlinear Schrödinger equation (BNLS). Let $T_{\rm c}$ be the collapse time, $r_{\rm \max}(t)$ the ring radius, $L(t)$ the scale, and $Q_{\mathrm B}$ the asymptotic ring profile.

Equal-rate blowup conjecture. The solution undergoes equal-rate collapse,

$$r_{\rm \max}(t)\sim r_0L(t),$$

and quasi-self-similar collapse with profile

$$\psi_{Q_{\mathrm B}}(t,r)=\frac{1}{L^{d/2}(t)}Q_{\mathrm B}\left(\frac{r-r_0L}{L}\right)e^{i\int_{s=0}^{t}\frac{1}{L^4(s)}\,ds}.$$

The blowup rate is exactly a quartic root:

$$L(t)\sim\kappa\sqrt[4]{T_{\rm c}-t},\qquad \kappa>0.$$

The source motivates this prediction from the lower bound $1/4$ and the square-root rate for critical NLS rings. It states the critical BNLS ring behavior as a conjecture and gives no resolution.