Poincaré-polynomial classification conjecture for rationally elliptic complex varieties

Let $X$ be a rationally elliptic complex algebraic variety, possibly singular, and let $P_X(t)$ denote its Poincaré polynomial. For an even-dimensional sphere $S^{2n_i}$ and a complex projective space $\mathbb{CP}^{m_j}$, write their Poincaré polynomials as $P_{S^{2n_i}}(t)$ and $P_{\mathbb{CP}^{m_j}}(t)$.

Poincaré-polynomial conjecture. There exist integers $k,s,n_i,m_j$ such that

$$P_X(t)=\prod_{i=1}^k P_{S^{2n_i}}(t)\times\prod_{j=1}^s P_{\mathbb{CP}^{m_j}}(t).$$

The source calls this a naive conjecture motivated by known examples. It does not give a resolution.