Is there some $\epsilon > 0$ such that there are infinitely many $n$ where all primes $p \le (2 + \epsilon) \log n$ divide $$\prod_{1 \le i \le \log n} (n + i)?$$

More generally, let $q(n, k)$ denote the least prime which does not divide $\prod_{1 \le i \le k}(n + i)$. This problem asks whether $q(n, \log n) \ge (2 + \epsilon) \log n$ infinitely often.