Universal quartic-root blowup conjecture for peak-type biharmonic NLS solutions

Let $\psi$ be a peak-type singular solution of the supercritical biharmonic nonlinear Schrödinger equation. Let $S_{\mathrm B}(\rho)$ be a radial self-similar profile satisfying

$$-\nu S_{\mathrm B}+i\frac{\kappa^4}{4}\left(\frac{2}{\sigma}S_{\mathrm B}+\rho S_{\mathrm B}'\right)-\Delta_\rho^2S_{\mathrm B}+|S_{\mathrm B}|^{2\sigma}S_{\mathrm B}=0,$$

with $S_{\mathrm B}'(0)=S_{\mathrm B}”'(0)=0$, $S_{\mathrm B}(\infty)=0$, and $H[S_{\mathrm B}]=0$. Universal quartic-root blowup conjecture. The collapsing core approaches $\psi_{S_{\mathrm B}}$; the profile is the unique admissible solution, $S_{\mathrm B}\neq R_{\mathrm B}$, and

$$L(t)\sim\kappa(T_{\rm c}-t)^{1/4},\qquad \kappa>0.$$

Moreover,

$$\kappa=\lim_{t\to T_{\rm c}}\frac{L(t)}{(T_{\rm c}-t)^{1/4}}=\kappa^{\rm admis.}(\sigma,d,\nu),$$

so the blowup coefficient is universal and independent of the initial condition. These are conjectural conclusions summarized by the paper; the admissible profile and its selection remain unproved.