The radial projection exceptional-set conjecture

Let $E\subset\mathbb{R}^d$ be a Borel set, let $d\geqslant 2$, and let $k=1,2,\dots,d-1$ satisfy

$$\dim_{\mathcal{H}} E\in(k-1,k].$$

Radial projection exceptional-set conjecture. One should have

$$\dim_{\mathcal{H}}\left\{x\in\mathbb{R}^d: \dim_{\mathcal{H}}\pi^x(E)<\dim_{\mathcal{H}}E\right\}\leqslant k.$$

The preceding theorem proves this type of bound when $\dim_{\mathcal{H}}E\in(d-2,d-1]$, and the conjectured estimate is stated to improve the Peres–Schlag bound. The endpoint case $\dim_{\mathcal{H}}E=d-1$ is already optimal for the preceding theorem, but the general assertion remains open.