Let $E$ be a ring spectrum and $F$ a connective spectrum. Then we have a convergent Atiyah-Hirzebruch spectral sequence $H_s(F,E_t) \Rightarrow E_{s+t}(F)$. Suppose now that $F$ is also a ring spectrum. Then does the multiplication on the $E^2$ page induce multiplications on all subsequent pages, and do they agree with the multiplication on $E_\ast(F)$?

Bizarrely, the only reference I can find for the multiplicative properties of the Atiyah-Hirzebruch spectral sequence is another MO question, and there is only treated the case of the cohomological AHSS where $F$ is a space, rather than the homological spectral sequence where $F$ is a ring spectrum.