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Claim. For every sufficiently large even $q$, every graph on $q^2+q+1$ vertices with $\frac12q(q+1)^2+1$ edges contains at least $q-1$ copies of $C_4$, and the paper characterizes the graphs with the fewest copies (He, Ma and Yang, Some extremal results on 4-cycles, J. Combin. Theory Ser. B 149 (2021), 92--108). The result first appeared as Theorem 1.5 of arXiv:1912.00986v1, Stability and supersaturation of $4$-cycles, for even $q\geq 10^{12}$: such a graph either contains at least $2q-3$ copies of $C_4$ or is an orthogonal polarity graph of order $q$ with one edge added, in which case it contains $q-1$, $q$ or $q+1$ copies. Later versions of that arXiv record became the authors' separate paper in CSIAM Trans. Appl. Math. 4 (2023), 74--128. When $q$ is a power of $2$, Füredi's bound and the polarity graph give $\mathrm{ex}(q^2+q+1;C_4)=\frac12q(q+1)^2$. Deleting edges only removes four-cycles, so every graph on $n=q^2+q+1$ vertices with more than $\mathrm{ex}(n;C_4)$ edges has at least $q-1=(1-o(1))\sqrt n$ copies of $C_4$, which is what Problem 60 asks at these orders. The v1 manuscript states this as Corollary 1.6: for $q=2^k$ with $k\geq 40$, the least number of copies of $C_4$ in a graph on $q^2+q+1$ vertices with $\mathrm{ex}(q^2+q+1;C_4)+1$ edges is exactly $q-1$, attained exactly by an orthogonal polarity graph of order $q$ with an edge added between two vertices of degree $q$.

Covers. The orders $n=q^2+q+1$ with $q$ a sufficiently large power of $2$. Nothing for other $n$; the problem stays open. The site's commentary credits the paper with the conjecture for every even $q$, but for even $q$ that is not a power of $2$ the value of $\mathrm{ex}(q^2+q+1;C_4)$ is not known to equal $\frac12q(q+1)^2$, so a graph with $\mathrm{ex}(n;C_4)+1$ edges need not have the theorem's edge count and the theorem does not reach the problem there; the formal-conjectures variant erdos_60.variants.he_ma_yang, linked from the problem page, states the result for powers of $2$ only.

Depends on. Nothing in this wiki.

Acceptance. Refereed: J. Combin. Theory Ser. B 149 (2021), 92--108. The site's commentary credits the paper, but the site labels the problem OPEN, so that credit is not reviewed evidence. No Lean proof of the result is recorded; the formal-conjectures variant carries sorry.