Consecutive integers free of certain prime factors

Let $n_k$ denote the least integer $n>2k$ such that $(n-k)(n-k+1)\cdots (n-1)$ is not divisible by any prime in the interval $(k,2k)$. Confirming a conjecture of Erdős, we prove that, for all sufficiently large $k$, $$n_k>e^{\frac{{\log }^{2}k}{20\log \log k}}\mathrm{.}$$