Rational points with large denominator on Erdős—Selfridge superelliptic curves

In 2016, Bennett and Siksek showed that if the Erdős—Selfridge curve $$(x+1)\cdots(x+k)=y^{\ell},\quad k\geq 3,\, \ell\mbox{ prime},$$ has a rational solution in $x$ and $y$, then $\ell\leq e^{3^k}$. In this paper, we show that if there exists a positive rational solution on the above curve, then either the denominator of the solution is large or $\ell\leq k$.