Finite-field radial projection exceptional-set conjecture

Let $E\subset\mathbb{F}_q^d$, where $q$ is a prime power, and let $k\in\{1,\ldots,d-1\}$ satisfy

$$q^{k-1}<|E|\leq q^k.$$

Finite-field radial projection exceptional-set conjecture.

$$\#\left\{y\in\mathbb{F}_q^d:|\pi^y(E)|<10^{-1}|E|\right\}\leq 10q^k.$$

This is proposed as a finite-field analogue of the Liu–Orponen conjecture; the paper proves related bounds in other size ranges but does not establish this estimate in general.