Asymptotic conjecture for the domatic number of almost regular graphs

Let $k$ be a positive integer and let $C\geq 1$ be a real number. A graph $G$ is $(k,C)$-regular if its minimum degree is $\delta(G)=k$ and its maximum degree satisfies $\Delta(G)\leq kC$. Let $f(k,C)$ be the minimum possible domatic number $dom(G)$ over all $(k,C)$-regular graphs.

Asymptotic domatic-number conjecture. For every fixed real number $C\geq 1$,

$$f(k,C)=\frac{k}{\ln k}(1+o_k(1)).$$

The preceding bounds show that $f(k,C)$ has order of magnitude $k/\ln k$ for fixed $C>1$, while the conjecture identifies the sharp leading constant and extends the proposed asymptotic to $C=1$, the regular-graph case. The source does not state a resolution, so the conjecture is recorded as open.