On products of consecutive integers

Let $p_k$ be the $k$-th prime number, and let $\nu_p(n)$ be the $p$-adic valuation of a positive integer $n$. Recently, Yang, Luca and Togbé proved that $\nu_p\big((p_k+1)(p_k+2)\cdots(p_{k+1}-1)\big)\leq \nu_p\left((\frac 12(p_{k+1}-1))!\right)$ for any integer $k\geq 5$ and any prime $p\leq \dfrac 12(p_{k+1}-1)$. In this paper, as a corollary, we prove that for any positive real number $\alpha$, there exists a positive integer $K_\alpha $ such that $\nu_p\big((p_k+1)(p_k+2)\cdots(p_{k+1}-1)\big)\leq \nu_p \left( \lfloor \alpha (p_{k+1}-1)\rfloor ! \right) $ for any integer $k\ge K_\alpha $ and any prime $p\leq \alpha (p_{k+1}-1)$.