List-recoverability conjecture for Reed–Solomon codes over prime fields

Let $\mathbb{F}_p$ be a prime field. Let $\ell \in \mathbb N$ and $\epsilon > 0$ be constants. For each $x \in \mathbb{F}_p$, let $S_x \subseteq \mathbb{F}_p$ satisfy $|S_x| \leq \ell$.

Reed–Solomon list-recoverability conjecture. The number of polynomials $P(X) \in \mathbb{F}_p[X]$ satisfying

$$\deg(P) \leq (1-\epsilon)p$$

and

$$P(x) \in S_x \quad\text{for every }x \in \mathbb{F}_p$$

is at most $p^{O_{\epsilon,\ell}(1)}$.

This conjecture asserts polynomially bounded list size for the relevant list-recovery instance of Reed–Solomon codes when the evaluation field is prime. The paper notes that results of Guruswami and Rudra imply that the prime-field assumption cannot simply be removed; the conjecture remains open in the stated generality.