Modified clique-free diameter conjecture

Let $k\geq 3$, let $\delta\geq \left\lceil\frac{3k}{2}\right\rceil-1$, and let $G$ be a connected graph of order $n$ and minimum degree at least $\delta$. The graph is assumed to be $K_{k+1}$-free; the source also gives the weaker alternative hypothesis that $G$ is $k$-colorable.

Modified diameter conjecture. Under either the $K_{k+1}$-free hypothesis, or the weaker $k$-colorable hypothesis, one should have

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

This modification is proposed after the even-clique case of the earlier conjecture is counterexampled. The paper presents it as a replacement conjecture; the $k$-colorable version is explicitly identified as weaker, and no resolution is given.