Structure of sets with nearly maximal Favard length

Let $E \subset B(1) \subset \mathbb R^{2}$ be an $\mathcal{H}^{1}$ measurable set with $\mathcal{H}^{1}(E) < \infty$, and let $L \subset \mathbb R^{2}$ be a line segment with $\mathcal{H}^{1}(L) = \mathcal{H}^{1}(E)$. It is not hard to see that $\mathrm{Fav}(E) \leq \mathrm{Fav}(L)$. We prove that in the case of near equality, that is, $$ \mathrm{Fav}(E) \geq \mathrm{Fav}(L) - δ, $$ the set $E$ can be covered by an $ε$-Lipschitz graph, up to a set of length $ε$. The dependence between $ε$ and $δ$ is polynomial: in fact, the conclusions hold with $ε= Cδ^{1/70}$ for an absolute constant $C > 0$.