A conjecture by Milnor state that if $G$ is a Lie group, then the map $B(G^{disc})\to BG$ sending the classifying space of the discrete group $G$ to the classifying space of the topological group $G$ induces an isomorphism on homology with $\mod p$ coefficients.

In chromatic homotopy theory, there are more "fields of characteristic p" than just finite fields, namely we have the Morava $K$-theories $K(p,n)$.

Question: Do we know "more of" the Milnor conjecture for those ring-spectra then for $\mathbb{F}_p$? (ultimately, but probably too ambitious, can it be proven for those ring spectra even though it is still open for $\mathbb{F}_p)$?

By "more of" I mean any progress that is special for this case and don't work for $\mathbb{F}_p$ coefficients.

A conjecture by Milnor state that if $G$ is a Lie group, then the map $B(G^{disc})\to BG$ sending the classifying space of $G$ endowed with the discrete topology to the classifying space of the topological group $G$ induces an isomorphism on homology with $\mod p$ coefficients.

In chromatic homotopy theory, there are more "fields of characteristic p" than just finite fields, namely we have the Morava $K$-theories $K(p,n)$.

Question: Do we know "more of" the Milnor conjecture for those ring-spectra then for $\mathbb{F}_p$? (ultimately, but probably too ambitious, can it be proven for those ring spectra even though it is still open for $\mathbb{F}_p)$?

By "more of" I mean any progress that is special for this case and don't work for $\mathbb{F}_p$ coefficients.