Besicovitch's almost-everywhere line intersection conjecture

Let $E\subset\mathbb{R}^2$ be $\mathcal{H}^1$ measurable and purely $1$-unrectifiable, with $\mathcal{H}^1(E)<\infty$. Besicovitch's almost-everywhere line intersection conjecture. For $\mathcal{H}^1$ almost all $a\in E$, almost all lines through $a$ meet $E$ only at $a$. The source presents this as a possible strengthening in the analysis of purely unrectifiable sets; it is stated as an unresolved possibility and no proof is given.