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

Taking $n$ to be the product of primes between $\log n$ and $(2 + o(1)) \log n$ gives an example where $$q(n, \log n) \ge (2 + o(1)) \log n.$$ Can one prove that $q(n, \log n) < (1 - \epsilon) (\log n)^2$ for all large $n$ and some $\epsilon > 0$?