Non-concentrated directions conjecture for projections of planar sets

Assume that $K \subset B(0,1) \subset \mathbb{R}^{2}$ is a set with $\mathcal{H}^{1}_{\infty}(K) \sim 1$. Let $E \subset S^{1}$ be a $\delta$-separated set of directions with cardinality $|E| \sim \delta^{-s}$, satisfying the non-concentration hypothesis

$$|E \cap B(x,t)| \lesssim t^{\kappa}|E|, \qquad x \in S^{1}, \: t \geq \delta,$$

for some $\kappa > 0$. Non-concentrated directions conjecture. There exists $\epsilon > 0$, depending only on $\kappa,s$, such that

$$N(\pi_{e}(K),\delta) \geq \delta^{-s - \epsilon}$$

for some $e \in E$. The conjecture asks whether quantitative non-concentration of the directions forces at least one projection to have more than the baseline number of $\delta$-intervals; the supplied material does not indicate whether this remains open or has been resolved.