Hilali's conjecture for mixed Hodge and homotopical Poincaré polynomials

Let $X$ be the space under consideration, with Poincaré polynomial $P_X(t)$ and homotopical Poincaré polynomial $P^{\pi}_X(t)$. At $(t,u,v)=(1,1,1)$, these satisfy

$$MH_X(1,1,1)=P_X(1)=1+\sum_{k\geqq 1}\dim H_k(X;\mathbb C),$$ $$MH^{\pi}_X(1,1,1)=P^{\pi}_X(1)=\sum_{k\geqq 2}\dim(\pi_k(X)\otimes\mathbb C).$$

Hilali conjecture. The homotopical Poincaré polynomial is bounded above by the Poincaré polynomial:

$$P^{\pi}_X(1)\leqq P_X(1),$$

i.e., $MH^{\pi}_X(1,1,1)\leqq MH_X(1,1,1)$. This has been proved for many classes of spaces, including smooth complex projective varieties and symplectic manifolds, but remains open in general.