Erdős–Pach–Pollack–Tuza conjecture on diameters of clique-free graphs

Let $r,\delta\geq2$ be fixed integers, and let $G$ be a connected graph of order $n$ and minimum degree $\delta$. Erdős–Pach–Pollack–Tuza conjecture.

(i) If $G$ is $K_{2r}$-free and $\delta$ is a multiple of $(r-1)(3r+2)$, then, as $n\to\infty$,

$$\operatorname{diam}(G)\leq\frac{2(r-1)(3r+2)}{2r^2-1}\cdot\frac{n}{\delta}+O(1)=\left(3-\frac{2}{2r-1}-\frac{1}{(2r-1)(2r^2-1)}\right)\frac{n}{\delta}+O(1).$$

(ii) If $G$ is $K_{2r+1}$-free and $\delta$ is a multiple of $3r-1$, then, as $n\to\infty$,

$$\operatorname{diam}(G)\leq\frac{3r-1}{r}\cdot\frac{n}{\delta}+O(1)=\left(3-\frac{2}{2r}\right)\frac{n}{\delta}+O(1).$$

The source says that counterexamples are known in a regime previously left open, so the displayed conjecture is not currently an open conjecture as stated; however, no precise resolution of both clauses is supplied here.