Let $X$ be a (finite, say — or maybe of finite type) 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).

• I don’t know many examples of complex-oriented $E$’s that don’t have homotopy concentrated in even degrees, so in practice the above suffices. But if this is true in general, I expect the proof the be illuminating.

• There’s an analog: if $E$ is a real-oriented cohomology theory, then the AHSS for $E^\ast(X)$ collapses at the $E_2$ page for any $X$. This is because Thom showed that the unoriented bordism spectrum $MO$ is an $H \mathbb F_2$-algebra. I’m not sure if there’s a more direct proof.