Let $\epsilon>0$ and $n$ be large depending on $\epsilon$. Is it true that for all $n^\epsilon<k\leq n^{1-\epsilon}$ the number of distinct prime divisors of $\binom{n}{k}$ is

$$(1+o(1))k\sum_{k<p<n}\frac{1}{p}?$$

Or perhaps even when $k \geq (\log n)^c$?