  We prove almost sure global existence and scattering for the energy-critical
nonlinear Schr\"odinger equation with randomized spherically symmetric initial
data in $H^s(\mathbb{R}^4)$ with $\frac56<s<1$. We were inspired to consider
this problem by the recent work of Dodson--L\"uhrmann--Mendelson, which treated
the analogous problem for the energy-critical wave equation.
