Liu–Orponen radial projection conjecture

Let $E\subset \mathbb{R}^d$ be a Borel set with $\dim_H(E)\in(k-1,k]$ for some $k\in\{0,1,\ldots,d-1\}$. Liu–Orponen's radial projection conjecture.

$$\dim_H\left\{y\in\mathbb{R}^d:\dim_H\pi^y(E)<\dim_H(E)\right\}\leq k.$$

This strengthens the exceptional-set estimate proved by Liu for sets of dimension at most $d-1$; the conjecture predicts that the exceptional set is controlled by the integer interval containing $\dim_H(E)$.