Let $E_\ast$ be a homology theory and $f: X \to Y$ a map of spaces. Suppose that $\Omega f: \Omega X \to \Omega Y$ is an $E_\ast$-equivalence. Then is $\Omega^2 f: \Omega^2 X \to \Omega^2 Y$ an $E_\ast$-equivalence?
This is true for ordinary integral homology, because if any map $\Omega X \to \Omega Y$ is an $H\mathbb Z$-equivalence, it is a homotopy equivalence by the homology Whitehead theorem (since loop spaces are simple). Alternatively (and more generally), a Serre spectral sequence argument shows that it also holds for $HR$-equivalences at least if $R$ is a PID (to use the universal coefficient theorem).
But is it true, say, for Morava K-theory?
(I have an abstract argument that it is true in general, but I don't trust it very much because it seems like the sort of fact which should hold for very classical reasons if it's true.)