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Claim. For $k=5$ there is an $n$ with $\prod_{0\le i\le 5}(n-i)\mid\binom{2n}{n}$, as Problem 396 asks: the least such $n$ is $3648841$, the term $a(5)$ of the OEIS entry A375077, which its extension line credits to Chai Wah Wu on 1 August 2024. A witness is checked through Kummer's theorem, as Stephan's page explains.

Covers. The instance $k=5$ of the question, with the answer yes by an explicit witness.

Depends on. No page of this wiki.

Standing. The entry is an edited database record, not a refereed publication, and the site's commentary on a problem it labels OPEN points to the entry without accepting a result. The claim stays claimed.