Flat-implies-Lipschitz conjecture with a small concentric hole

Let $u$ be a classical solution to the one-phase free-boundary problem in the punctured ball $\mathbb{B}_1\setminus\mathbb{B}_\delta$, with contractible positive and zero phases. Flat-implies-Lipschitz conjecture with a spherical hole. There are absolute constants $\delta_0>0$, $\epsilon>0$, and $C$ such that, whenever $\delta\le\delta_0$ and, for every $x=(x_1,x_2,x_3)\in\mathbb{B}_1\setminus\mathbb{B}_\delta$,

$$x_3>\epsilon\implies u(x)>0,\qquad x_3<-\epsilon\implies u(x)=0,$$

the free boundary satisfies

$$F(u)\cap\mathbb{B}_{1/C}\cap\{|x|>C\delta\}=\{x\in\mathbb{B}_{1/C}:x_3=g(x_1,x_2),\ |x|>C\delta\}$$

for some $g:\mathbb{R}^2\to\mathbb{R}$ with $|\nabla g|\le 1/100$.

This is the spherical-hole analogue of the preceding flat-implies-Lipschitz formulation. The source says that such solutions should resemble a half-space solution or one half of the solution constructed by Liu and collaborators.