Let $E$ be a spectrum and let $X$ be a space or connective spectrum. Then the cohomological Atiyah-Hirzebruch spectral sequence is of the form:
$H^s(X,E^t) \Rightarrow E^{s+t}(X)$
This is a half-plane spectral sequence with entering differentials. If $E' \to E$ is a map of spectra, then I believe I get a map of spectral sequences to this from the analogous spectral sequence for $(E')^\ast(X)$. In particular, let $E' = \tau_{\geq k} E$ be a connective cover. Then for degree reasons, any differential $d_u^{k,t}$ vanishes in the $E'$-AHSS. By mapping forward along the map of spectral sequences, this seems to imply that $d_u^{k,t}$ also vanishes in the $E$-AHSS. But since this holds for all $k \in \mathbb Z$, we seem to have concluded that the $E$-AHSS collapses, which can't be right at all.
Where did this argument go wrong?