Green-function decay and boundary regularity conjecture

Let $\Omega\subset\mathbb R^n$ be a one-sided NTA domain whose Green function with some pole $p$ satisfies, for some $\delta>0$ and constants $c_1,c_2>0$, bounds of the form

$$c_1\operatorname{dist}(z,\partial\Omega)^\delta\leq G(z,p)\leq c_2\operatorname{dist}(z,\partial\Omega)^\delta$$

for all $z\in\Omega$ sufficiently close to $\partial\Omega$. Green-function decay conjecture. Necessarily $\delta=1$, and $\partial\Omega$ has a certain weak smoothness, at least being rectifiable. The statement is presented as an easier conjecture related to the preceding boundary-regularity questions, but no general proof is supplied.