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Claim. For $k=1,2,3,4$ 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 $2$, $2480$, $8178$ and $45153$. They are the first four terms of the OEIS entry A375077, "Smallest k such that Product_{i=0..n} (k-i) divides C(2*k,k)" (the entry's $n$ is the problem's $k$ and its $k$ the problem's $n$), which Ralf Stephan authored on 29 July 2024 with these four terms; the site's commentary points to the entry for the least $n$ of each $k$. A witness is checked through Kummer's theorem: for every prime $p$ the sum of the exponents of $p$ in $n,n-1,\ldots,n-k$ is at most the number of carries when $n$ is added to itself in base $p$, which is the exponent of $p$ in $\binom{2n}{n}$.

Covers. The instances $k=1,2,3,4$ of the question, each with the answer yes by an explicit witness. The entry's later terms are recorded on the pages of their contributors; the question for every $k$ is open.

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.