  In his study of Halperin's toral-rank conjecture, M. R. Hilali conjectured
that for any simply connected rationally elliptic space $X$, one must have
$dim\pi_*(X)\otimes \mathbb{Q} \leq dimH^*(X,\mathbb{Q})$. Let $(\Lambda V, d)$
denote a Sullivan minimal model of $X$ and $d_k$ the first non-zero homogeneous
part of the differential $d$. In this paper, we use spectral sequence arguments
to prove that if $(\Lambda V, d_k)$ is elliptic, then, there is no gaps in the
$E_{\infty}$ term of the Milnor-Moore spectral sequence of $X$. Consequently,
we confirm the Hilali conjecture when $V = V^{odd}$ or else when $k\geq 3$ and
$(\Lambda V, d_k)$ is elliptic.
