Product representations of perfect powers

Let $\rho_k(N)$ denote the maximum size of a set $A\subseteq \{1,2,\dots,N\}$ such that no product of $k$ distinct elements of $A$ is a perfect $d$-th power. In this short note, we prove that $\rho _d(N)=\sum\limits_{k=1}^{d-1}\pi\left( \frac{N}{k} \right) +O_d(\pi (N^{1/2}))$, furthermore, for prime power $d$ and sufficiently large $N$ we have $\rho _d(N)=\sum\limits_{k=1}^{d-1}\pi\left( \frac{N}{k} \right)$. This answers a question of Verstra\"ete.