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Claim. Éva Czabarka, Inne Singgih and László A. Székely refute part (i) of the statement of Problem 612. Theorem 6 of the arXiv preprint (its Section 3, pp. 7--8): for every $r\ge2$, every $\delta\ge2r-2$ and every positive integer $p$ there is a connected $(2r-1)$-colorable, hence $K_{2r}$-free, graph $G_{r,\delta,p}$ of minimum degree $\delta$, order $n=p((2r-1)\delta+2r-3)+2$ and diameter

$$\frac{(6r-5)\,n}{(2r-1)\delta+2r-3}+O(1).$$

This ratio exceeds the conjectured $\frac{2(r-1)(3r+2)}{(2r^2-1)\delta}$ exactly when $\delta>2(r-1)(3r+2)(2r-3)$, so part (i) fails for every $r\ge2$ and every $\delta$ in that range with $(r-1)(3r+2)\mid\delta$. The paper leaves the window $(r-1)(3r+2)\le\delta\le2(r-1)(3r+2)(2r-3)$ of part (i) open (its p. 2) and proposes an amended conjecture, which is not the problem.

Covers. Part (i), in full: it is a conjunction over all $r\ge2$ and all admissible $\delta$, so a single false instance refutes it, and this page settles it negatively. Part (ii), the question for $K_{2r+1}$-free graphs, is untouched by this construction and stays open on the problem page.

Acceptance. The counterexample paper is refereed: J. Combin. Theory Ser. B 151 (2021), 38--45, issued November 2021. Czabarka, Smith and Székely restate the counterexample and its open window on p. 2 of their refereed paper in J. Graph Theory 102 (2023), 262--270, and the site's own commentary records this paper as a disproof for the case of $K_{2r}$-free graphs, that is of part (i), while the site's label stays OPEN, which the problem page reads as leaving part (ii) open. The theorem is stated from the preprint; the published article is not held, so its theorem numbering is unchecked, and the proof is not independently checked in this corpus. The source card is czabarka_2021_counterexamples_conjecture_erdos_pach_pollack_tuza, which records the preprint and the Electron. J. Combin. article that took the preprint's $k$-colorable results and holds neither file.