Quantitative Besicovitch projection conjecture for AD-regular planar sets

Let $s\in (0,1)$ and $C_0\in (1,\infty)$. Let $E\subset \mathbb{R}^2$ be a bounded AD-regular set with constant $C_0$, meaning that $E$ is closed and, for every $x\in E$ and $0<r<\operatorname{diam}(E)$,

$$C_0^{-1}r\leq\mathcal{H}^1(E\cap B(x,r))\leq C_0r.$$

Write $\operatorname{Fav}(E)$ for the Favard length of $E$, and let $\operatorname{Lip}(\Gamma)$ denote the Lipschitz constant of a Lipschitz graph $\Gamma$. Quantitative Besicovitch projection conjecture. If

$$\operatorname{Fav}(E)\geq s\operatorname{diam}(E),$$

then there exists a Lipschitz graph $\Gamma\subset\mathbb{R}^2$ such that

$$\operatorname{Lip}(\Gamma)\lesssim_{s,C_0}1$$

and

$$\mathcal{H}^1(\Gamma\cap E)\gtrsim_{s,C_0}\mathcal{H}^1(E).$$

This would provide a quantitative counterpart to Besicovitch's qualitative projection theorem, with both the Lipschitz constant and the length of the intersection controlled in terms of $s$ and $C_0$. The statement is presented as a conjectural quantitative result in the source, and its resolution is not supplied here.