OPEN

Is it true that for every $n$ there is a $k$ such that
\[\prod_{1\leq i\leq k}(n+i) \mid \prod_{1\leq i\leq k}(n+k+i)?\]

Asked by Erdős and Straus.
For example when $n=2$ we have $k=4$:
\[3\cdot 4\cdot 5\cdot 6 \mid 7\cdot 8\cdot 9\cdot 10.\]
No example was known for $n\geq 10$.