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Claim. Write $f^{(r)}(n;s,k)$ for the largest number of edges of an $r$-uniform hypergraph on $n$ vertices containing no $k$ edges on at most $s$ vertices, the paper's notation. Unlike the function of Brown, Erdős and Sós, which is the least number of edges forcing such a configuration, it is a maximum, so the site's extremal number is $\mathrm{ex}_3(n,\mathcal F_k)=f^{(3)}(n;k,k-2)$. Theorems 1.1 and 1.2 of Glock, Kim, Lichev, Pikhurko and Sun state that for every $r\ge3$ the limits of $n^{-2}f^{(r)}(n;5r-8,5)$ and of $n^{-2}f^{(r)}(n;7r-12,7)$ both equal $1/(r^2-r-1)$, and Theorem 1.3 states that $n^{-2}f^{(3)}(n;8,6)\to61/330$. At $r=3$ the first two concern $f^{(3)}(n;7,5)$ and $f^{(3)}(n;9,7)$, with limit $1/5$. Hence

$$\mathrm{ex}_3(n,\mathcal F_7)=\Bigl(\frac15+o(1)\Bigr)n^2,\qquad \mathrm{ex}_3(n,\mathcal F_8)=\Bigl(\frac{61}{330}+o(1)\Bigr)n^2,\qquad \mathrm{ex}_3(n,\mathcal F_9)=\Bigl(\frac15+o(1)\Bigr)n^2,$$

and none of these is $(1/6+o(1))n^2$, since $1/6=55/330$: the displayed asymptotic fails at $k=7$, $8$ and $9$ with $\mathcal F_k$ the single family the site's wording defines, so the site's wording, an assertion about every $k\ge5$, is false. The transfer from the theorems to the site's wording is the identification of the forbidden families: a $3$-graph contains a member of $\mathcal F_k$ exactly when some $k-2$ of its edges span at most $k$ vertices. The earlier refutations at $k=5$ and $k=6$ are on Glock's page and the (6,4) page, and a refutation at $k=10$ is on the (10,8) page.

Why it is rejected. The results are correct, but they answer the site's wording, the single families $\mathcal F_7$, $\mathcal F_8$ and $\mathcal F_9$, not the corrected Statement of Problem 1076, whose family is cumulative: under the corrected Statement a $3$-graph avoiding $\mathcal F_4\cup\dots\cup\mathcal F_k$ is linear, so the theorems say nothing against it at any of these $k$, and the page does not count toward the problem's standing. The problem page's Notes credit the results.

Acceptance. Refereed: S. Glock, J. Kim, L. Lichev, O. Pikhurko and S. Sun, On the $(k+2,k)$-problem of Brown, Erdős, and Sós for $k=5,6,7$, Canad. J. Math. 78 (2026), no. 5, 1566–1608, published online 6 January 2025 after the arXiv posting of 7 March 2024. The abstract presents the results as the cases $k=5,6,7$ of Brown, Erdős and Sós's conjecture that the limit of $f^{(3)}(n;k+2,k)/n^2$ exists, after the cases $k=2$ (Brown, Erdős and Sós), $k=3$ (Glock) and $k=4$ (Glock, Joos, Kim, Kühn, Lichev and Pikhurko), the existence of the limit for every $k$ having been proved by Delcourt and Postle. The site does not cite the paper on this problem and its curator makes no statement about it; the proof is unreviewed, and no formalization is known.