Almost sure scattering for the energy-critical NLS with radial data below $H^1({\mathbb{R}}^4)$

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 $\frac{5}{6}<s<1$. 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.