Gradient stability for the Sobolev inequality: the case $p\geq 2$

We prove a strong form of the quantitative Sobolev inequality in $\mathbb{R}^n$ for $p\geq 2$, where the deficit of a function $u\in \dot W^{1,p}$ controls $\| \nabla u -\nabla v\|_{L^p}$ for an extremal function $v$ in the Sobolev inequality.