Hilali conjecture on rational homotopy and homology dimensions

Let $X$ be a simply connected rationally elliptic space. Write $\boldsymbol{\pi_*(X)}$ for its rational homotopy groups and $H_*(X;\mathbb Q)$ for its rational homology groups. Hilali conjecture.

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

The conjecture compares the total dimensions of the rational homotopy and homology of an elliptic space. It remains open, although it is known for several classes of spaces, including elliptic spaces of pure type, H-spaces, nilmanifolds, symplectic and cosymplectic manifolds, formal spaces, and certain hyperelliptic spaces.