Given an omega spectrum $E$, there is a type of chern character map given by its rationalization $$r:E\to E\wedge M\mathbb{R}\;,$$ where $M\mathbb{R}$ denotes a Moore spectrum. The cofiber of the map $$\mathbb{S}\to M\mathbb{R}$$ can be identified with an $M\mathbb{R}/\mathbb{Z}$ and smashing sequence on the left by $E$ yields a Bockstein sequence $$E\overset{r}{\to} E\wedge M\mathbb{R} \to E \wedge M\mathbb{R}/\mathbb{Z}\overset{\beta}{\to}\Sigma E\;.$$

Both $r$ and $\beta$ induce morphisms of corresponding Atiyah-Hirzebruch spectral sequences. Although $\beta$ is usually trivial in practice (its $0$ when the coefficients are concentrated in even degrees for example). If I assume that $\pi_*(E)$ is concentrated in even degrees and is (say free) in those degrees, then there is also a Bockstien morphism corresponding to the sequence $$0\to \pi_*(E)\to \pi_*(E)\otimes \mathbb{R} \to \pi_*(E)\otimes \mathbb{R}/\mathbb{Z}\to 0,$$ induces a map on the $E_2$ page of the corresponding theories. I was hoping to show that this map commutes with the differentials and that the induced map on higher pages commutes (essentially that it defines a morphism of spectral sequences). Does anyone know if this is true, or know of a reference where this is discussed?