Boundary convexity conjecture for the Beer index

Let $S\subseteq\mathbb{R}^2$ be a measurable set, and let $\operatorname{b}(S)$ denote its Beer index and $\lambda$ planar Lebesgue measure. Boundary convexity conjecture. For every $\varepsilon>0$, there is a $\delta>0$ such that if

$$\operatorname{b}(S)\geq\varepsilon,$$

then there is a bounded convex set $C\subseteq\mathbb{R}^2$ satisfying

$$\lambda(C)\geq\delta\lambda(S)\qquad\text{and}\qquad\partial C\subseteq S.$$

The conjecture concerns arbitrary planar sets, including sets with no convex subset of positive measure. It is known for p-componentwise simply connected sets, where $\delta$ can be chosen as a constant multiple of $\varepsilon$; the general assertion remains open.