Let $X$ be a (finite, say) spectrum with even cells (in other words, $H_\ast(X;\mathbb Z)$ is free and concentrated in even degrees). Let $E^\ast$ be a complex-oriented cohomology theory. Consider the Atiyah-Hirzebruch spectral sequence $H^p(X,E_q) \implies E^{p+q}(X)$.

Question: Does this spectral sequence always collapse at the $E_2$ page?

Notes:

• The answer is yes if $X = \mathbb C \mathbb P^n$, $BU(n)$, $MU$, etc.

• The answer is yes if $E$ has homotopy concentrated in even degrees (this is how one shows that such an $E$ is complex-orientable).