The spherically symmetric scattering conjecture for mass-critical NLS

Let $u$ be a spherically symmetric solution of the mass-critical nonlinear Schrödinger equation on ${\mathbf{R}}^d$. Let $M(u)$ denote its mass, $S(u)$ its scattering size, and define $A(m)_{\operatorname{rad}}$ as the supremum of $S(u)$ over spherically symmetric solutions with $M(u)\leq m$. Let $m_{0,\operatorname{rad}}$ be the unique threshold such that $A(m)_{\operatorname{rad}}$ is finite for $m<m_{0,\operatorname{rad}}$ and infinite for $m\geq m_{0,\operatorname{rad}}$. Let $Q$ be the ground state satisfying

$$\Delta Q+Q^{1+4/d}=Q.$$

Spherically symmetric scattering conjecture. In the defocusing case ($\mu=+1$) we have $m_{0,\operatorname{rad}}=+\infty$, while in the focusing case ($\mu=-1$) we have $m_{0,\operatorname{rad}}=M(Q)$. The radial restriction is expected to simplify the concentration-compactness problem, while the ground state remains the natural focusing threshold. The statement is presented as the spherically symmetric counterpart of the preceding conjecture; its resolution is not specified in the supplied material.