I posed this question to another user, but we weren't able neither to make a precise guess, nor to give a precise proof. Is it a "not-so-well" well-known fact?

Consider a connected non-finite CW complex $X$; denote its Poincare' series as $p_X(T)=a_0+a_1T+\dots + a_nT^n+\dots$

Since $X$ is connected, the series $p_X(T)$ is an invertible element in $\mathbb Z[\![T]\!]$: what is the geometric meaning of its inverse series $q(T)$? Can it arise as the series of a space $Y$? Which is the relation between $X$ and $Y$?