General blowup conjecture for the L^2-critical boson star equation

Suppose that $u \in C^0([0,T);H^{1/2}(\mathbb{R}^3))$ solves the boson star equation with $m\geqslant 0$ and blows up at a finite time $0<T<+\infty$. Let $u^*\in L^2(\mathbb{R}^3)$ and $\mu\in\mathcal M(\mathbb{R}^3)$ be the limiting profile and blowup measure from the nonradial blowup theorem. General blowup conjecture. There exist finitely many points $\\{x_1,\ldots,x_L\\}\subset\mathbb{R}^3$, with $1\leqslant L\leqslant \frac{\int |u_0|^2}{M_{\rm c}}$, such that, as $t\to T^-$,

$$u u(t,\cdot)\to u_* \text{ strongly in }L^2\left(\mathbb{R}^3\setminus\bigcup_{1\leqslant i\leqslant L}B(x_i,R)\right)\text{ for all }R>0,$$

and

$$|u(t,\cdot)|^2\rightharpoonup \mu=\sum_{1\leqslant i\leqslant L}M_i\delta_{x=x_i}+|u_*|^2$$

weakly in $\mathcal M(\mathbb{R}^3)$, with $M_i\geqslant M_{\rm c}$. This conjecture describes the singular part of the blowup measure as finitely many point masses, each carrying at least the critical mass, while the solution converges strongly away from the singular points. It is inspired by the analogous Merle–Raphaël conjecture for the $L^2$-critical nonlinear Schrödinger equation; the supplied text does not state whether the conjecture has been resolved.