Erdős–Selfridge conjecture on prime divisors of consecutive products

Let $k\geq 4$, let $n$ be an integer, and let $p^{(k)}$ denote the smallest prime greater than or equal to $k$. The product of $k$ consecutive integers is

$$(n+1)\cdots(n+k).$$

Erdős–Selfridge conjecture. If $n+k\geq p^{(k)}$, then there is a prime greater than $k$ which divides $(n+1)\cdots(n+k)$ to the first power.

This conjecture is attributed in the source to Erdős and Selfridge and is stated because it implies the index divisor free prime conjecture when $k\geq 5$ and $k$ is not prime. Its resolution status is not specified in the supplied text.