A quadratic bound for Jacobsthal's function

Let $h(k)$ be the least integer such that every interval of $h(k)$ consecutive integers contains an integer coprime to any prescribed positive integer having at most $k$ distinct prime divisors. We prove $h(k)\ll k^2/(\log\log(3k))^2$, giving an affirmative answer to Jacobsthal's quadratic-bound question.