Hilali's conjecture as the contractible-target case

Let $X$ be a simply connected elliptic space, and define

$$P_X(t)=\sum_{i\geq 0}t^i\dim H_i(X;\mathbb Q),\qquad P_X^{\pi}(t)=\sum_{i\geq 1}t^i\dim(\pi_i(X)\otimes\mathbb Q).$$

Hilali's conjecture. The inequality

$$P_X^{\pi}(1)\leq P_X(1),$$

that is,

$$\sum_{i\geq 2}\dim(\pi_i(X)\otimes\mathbb Q)\leq 1+\sum_{i\geq 2}\dim H_i(X;\mathbb Q),$$

should hold. This is identified in the paper as the special case of the relative Hilali conjecture when the target is a point or contractible; the source also discusses the broader Hilali conjecture above.