The Hilali conjecture on product of spaces

The Hilali conjecture claims that a simply connected rationally elliptic space $X$ satisfies the inequality $\operatorname{dim} (\pi_*(X)\otimes \mathbb Q ) \leqq \operatorname{dim} H_*(X;\mathbb Q )$. In this paper we show that for any such space $X$ there exists a positive integer $n_0$ such that for any $n \geqq n_0$ the \emph{strict inequality} $\operatorname{dim} (\pi_*(X^n)\otimes \mathbb Q ) < \operatorname{dim} H_*(X^n;\mathbb Q )$ holds, where $X^{n}$ is the product of $n$ copies of $X$.