Consider a fibration $F\to E\to B$ where $H^i(F;\mathbb{Q})$ and $H^i(B;\mathbb{Q})$ are finite-dimensional, and they vanish for $i\gg 0$, and $B$ is connected. However, we do not assume that $B$ is finite or simply connected; it might be the classifying space of an acyclic group, for example. (See https://arxiv.org/pdf/1006.4009.pdf for some general discussion of acyclic groups.) Is it still true that $H^*(E;\mathbb{Q})$ has finite total dimension, and that $\chi(E)=\chi(F)\chi(B)$?
There was an earlier question Multiplicativity of Euler characteristic for non-orientable fibrations which asked about this under the assumption that $B$ is a finite complex. In that case there is a simple argument using induction over the cells of $B$. Later Mike Shulman (who asked that question) added a comment linking to his paper The multiplicativity of fixed point invariants with Kate Ponto, which uses methods akin to parameterised stable homotopy. Using the fact that there are finite complexes $B',F'$ and rational equivalences $B'\to\Sigma^2B$ and $F'\to\Sigma^2F$, it might be possible to deduce a positive answer to my question from the results in that paper. However, significant work would be needed to fill in the details, and I wonder whether there is a more direct argument.
