The strong pinned radial-projection conjecture for planar sets

Let $E$ and $F$ be Borel sets in the plane with $\dim_{\mathcal{H}}E>0$ and $\dim_{\mathcal{H}}F>0$, and suppose that $F$ is not contained in a line. For $y$ in the plane, let $\pi^y(E)$ denote the radial projection of $E$ from $y$. Strong pinned radial-projection conjecture. There exists $y\in F$ such that

$$\dim_{\mathcal{H}}\pi^y(E)=\min\\{\dim_{\mathcal{H}}E,1\\}.$$

The paper calls this a stronger guess than both the exceptional-set conjecture above and Orponen's direction-set conjecture. It is presented as open, and the paper's results are described as progress toward the related pinned statement.