  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 $X$.
