Can one show the stronger version with \[\omega(n-k) < \frac{\log k}{\log\log k}+O(1)\] is false?

OPEN

Let $\epsilon>0$ and $\omega(n)$ count the number of distinct prime factors of $n$. Are there infinitely many values of $n$ such that
\[\omega(n-k) < (1+\epsilon)\frac{\log k}{\log\log k}\]
for all $k<n$ which are sufficiently large depending on $\epsilon$ only?

Can one show the stronger version with \[\omega(n-k) < \frac{\log k}{\log\log k}+O(1)\] is false?

One can ask similar questions for $\Omega$, the number of prime factors with multiplicity, where $\log k/\log\log k$ is replaced by $\log_2k$.