Realizability of the Adams-Novikov spectral sequence for formal $A$-modules

We show that the formal $A$-module Adams-Novikov spectral sequence of Ravenel does not naturally arise from a filtration on a map of spectra by examining the case $A = \mathbb{Z}[i]$. We also prove that when $A$ is the ring of integers in a nontrivial extension of $\mathbb{Q}_p$, the map $(L,W) \to (L_A,W_A)$ of Hopf algebroids, classifying formal groups and formal $A$-modules respectively, does not arise from compatible maps of $E_\infty$-ring spectra $(MU,MU \wedge MU) \to (R,S)$.