Czabarka–Singgih–Székely diameter conjecture for graphs with bounded clique number

Let $k\geq3$, let $\delta\geq\left\lceil\frac{3k}{2}\right\rceil-1$, and let $G$ be a connected graph of order $n$ with minimum degree at least $\delta$. Write $\omega(G)$ for its clique number and $\chi(G)$ for its chromatic number.

Czabarka–Singgih–Székely conjecture. If $\omega(G)\leq k$, then

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

A weaker version assumes $\chi(G)\leq k$ instead of $\omega(G)\leq k$. This updates the earlier conjecture after counterexamples to its even-clique case, but its resolution is not specified in the supplied text.