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Claim. For $k=8,9,10,11$ and $k=15$ 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

$$339949252,\quad 1019547844,\quad 17609764994,\quad 1070858041585$$

for $k=8,\ldots,11$ and $2394789405254721$ for $k=15$. They are the terms $a(8)$ to $a(11)$ and $a(15)$ of the OEIS entry A375077, whose extension lines credit them to Sharvil Kesarwani on 18 March 2026 and 14 July 2026. Kesarwani's posts in the site's discussion thread, from 25 March 2026 on, describe the optimizations of their search program, which a third party ran to find the $k=14$ term recorded on the joint page. A witness is checked through Kummer's theorem, as Stephan's page explains; the minimality of the terms $a(8)$ to $a(13)$ is separately certified by the exhaustive search recorded on Dehorty's page.

Covers. The instances $k=8,9,10,11$ and $k=15$ 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.