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Claim. A. Grebennikov, A. Sagdeev, A. Semchankau and A. Vasilevskii, On the sequence $n!\bmod p$, Rev. Mat. Iberoam. 40 (2024), no. 2, 637-648 (arXiv:2204.01153, first version 3 April 2022, which already contains both results below). Writing $\mathcal A(p)=\{i!\bmod p:i\in[p-1]\}$, the set $A_p$ of Problem 478, the paper proves (Theorem 1)

$$|\mathcal A(p)\mathcal A(p)|\ge p+O\big(p^{13/14}(\log p)^{4/7}\big),$$

and deduces (Corollary 1)

$$|\mathcal A(p)|\ge(\sqrt2+o(1))\sqrt p,$$

improving García's constant $\sqrt{41/24}$. The paper's source card is Grebennikov, Sagdeev, Semchankau and Vasilevskii 2024.

Covers. The lower bound $|A_p|\ge(\sqrt2+o(1))\sqrt p$ only. It does not prove $|A_p|\gg p$, nor the asymptotic $|A_p|\sim(1-1/e)p$ the problem asks for, and the paper says that Erdős's conjecture $|A_p|<p-2$ remains open.

Depends on. Nothing in this wiki; the claim is the cited paper's theorem.

Acceptance. Refereed: Revista Matemática Iberoamericana 40 (2024). The site labels the problem OPEN, and its remarks credit this paper with the best known lower bound; that credit is not acceptance.