Czabarka–Dankelmann–Székely conjecture on diameters of clique-free graphs

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 $\delta$. A graph is $K_{k+1}$-free if it contains no complete subgraph on $k+1$ vertices; the source also mentions the stronger hypothesis that $G$ is $k$-colorable.

Czabarka–Dankelmann–Székely conjecture. For every such $G$,

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

This modified conjecture was proposed after the even-clique case of the earlier conjecture was disproved. It removes the parity distinction between excluded complete subgraphs; the source does not report a resolution.