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Claim. For $k=6$ and $k=7$ there is an $n$ with $\prod_{0\le i\le k}(n-i)\mid\binom{2n}{n}$, as Problem 396 asks: the least such $n$ are $7979090$ and $101130029$, the terms $a(6)$ and $a(7)$ of the OEIS entry A375077, which its extension line credits to Max Alekseyev on 25 February 2025. A witness is checked through Kummer's theorem, as Stephan's page explains.

Covers. The instances $k=6$ and $k=7$ of the question, each 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.