Hilali conjecture for rationally elliptic spaces

A simply connected topological space $X$ is rationally elliptic if

$$\dim\left(\pi_*(X)\otimes\mathbb{Q}\right)<\infty$$

and

$$\dim H_*(X;\mathbb{Q})<\infty.$$

Hilali conjecture. If $X$ is a rationally elliptic space, then

$$\dim\left(\pi_*(X)\otimes\mathbb{Q}\right)\leq \dim H_*(X;\mathbb{Q}).$$

This conjecture compares the total rational homotopy rank with the total rational homology rank; it is known for several classes and in bounded formal dimensions, but remains open in general.