Uniform Rectifiability and harmonic measure IV: Ahlfors regularity plus Poisson kernels in $L^p$ implies uniform rectifiability

Let $E\subset {\mathbb{R}}^{n+1}$, $n\ge 2$, be an Ahlfors-David regular set of dimension $n$. We show that the weak-$A_{\mathrm{\infty }}$ property of harmonic measure, for the open set $\mathrm{\Omega }:={\mathbb{R}}^{n+1}\setminus E$, implies uniform rectifiability of $E$.