Is it true that, for every $r$, there is a $k$ such that if $I_1,\ldots,I_r$ are disjoint intervals of consecutive integers, all of length at least $k$, then $$\prod_{1\leq i\leq r}\prod_{m\in I_i}m$$ is not a perfect power?

Erdos and Selfridge [ErSe75] proved that the product of consecutive integers is never a power (establishing the case $r=1$).

Theorem 1 from [ErSe75].

It is implied from erdos_930.variants.consecutive_strong.

[ErSe75] Erdős, P. and Selfridge, J. L., The product of consecutive integers is never a power. Illinois J. Math. (1975), 292-301.