Improved exponent bounds for Erdős--Selfridge curves

Given integers $k,\ell \geq 2$, define the Erdős--Selfridge curve \begin{align*} \mathcal{C}_{k,\ell} : y^\ell = x(x+1)\cdots (x+k-1). \end{align*} Bennett and Siksek proved that if $(x,y)$ is a rational point on $\mathcal{C}_{k,\ell}$ with $\ell$ a prime and $y \neq 0$, then $\ell$ is bounded (doub