Quantitative Besicovitch conjecture for planar sets

Let $\sigma>\frac{1}{2}$. The notation $\mathcal{H}^1$ denotes one-dimensional Hausdorff measure, $\operatorname{diam}(U)$ the diameter of $U$, and $B_r(x)$ the ball of radius $r$ centered at $x$.

Quantitative Besicovitch conjecture. There exist positive constants $\varepsilon$ and $\delta$ such that, whenever $E\subset\mathbb R^2$ is compact and

$$\mathcal{H}^1(E\cap U)\leq (1+\varepsilon)\operatorname{diam}(U)$$

for every $U$ with $\operatorname{diam}(U)\leq 1$, and

$$\mathcal{H}^1(E\cap B_r(x))\geq 2\sigma r$$

for every $x\in E$ and every $r\leq 1$, any two connected components of $E$ are at least $\delta$ apart. Consequently, provided $\varepsilon$ is possibly smaller, $E$ is the union of finitely many disjoint embedded closed loops.

The paper proposes this as a quantitative version of Besicovitch's conjecture. It concerns uniform separation and structural finiteness under quantitative upper and lower measure bounds; its resolution is not given in the source.