This question is about the state of current knowledge regarding Voevodsky's algebraic cobordism of a point $\mathrm{MGL}^{*,*}(\mathrm{Spec}\,k)$. That the geometric diagonal $\mathrm{MGL}^{2*,*}(\mathrm{Spec}\,k)$ is isomorphic to the Lazard ring $\mathbb{L}$ if $char(k)=0$ has been shown by Levine using the Hopkins-Morel isomorphism. For positive characteristic fields, Hoyois showed the same isomorphism holds after inverting the characteristic exponential. It is also well-known that $\mathrm{MGL}^{n,n}(\mathrm{Spec}\,k)$ is identified with the Milnor $K$-theory of a field.

Spitzweck, in Algebraic Cobordism in mixed characteristic, section 7, computed some of the homotopy groups $\pi_{p,q}\mathrm{MGL}_S$ of the algebraic cobordism spectrum in the stable homotopy category $SH(S)$ over $S$, $S$ being the spectrum of a Dedekind domain of mixed characteristic. For instance, he described the cases $(p,q)=(2n+1,n)$, $(n+1,n)$. My question is:

- Are similar descriptions already known in the case $S=\mathrm{Spec}\,k$?
- For $S=\mathrm{Spec}\,k$, is it known that $\mathrm{MGL}^{2n+i,n}(\mathrm{Spec}\,k)$ vanishes for $i\geq 1$.