Algebraic cobordism (of a point) outside the geometric diagonal

Question

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:

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

Answer 1

The answer to both question is yes, provided you invert $p$ in characteristic $p$ (though conjecturally this is not necessary). In fact, as far as I can see, all of Spitzweck's computations apply to fields as well. He only assumes mixed characteristic because some of the arguments are more difficult in that case (e.g. Corollary 5.9). So for example $\mathrm{MGL}^{2n-1,n}(\operatorname{Spec} k) = k^\times \otimes L_{1-n}$ and $\mathrm{MGL}^{n-1,n}(\operatorname{Spec} k)$ is an extension of $H^{n-1,n}(\operatorname{Spec} k)$ by a quotient of $K_{n+1}^M(k)$.

For the vanishing see also this question: The vanishing of $MGL^{2n+i,n}(X)$; do spectra of smooth projective varieties generate $SH_{l}$?

It's worth noting that the vanishing of $\mathrm{MGL}^{p,q}(\operatorname{Spec}k)$ for $p>q$ and the computation $\mathrm{MGL}^{n,n}(\operatorname{Spec}k)=K_n^M(k)$ hold without inverting the characteristic (they come from Morel's results about the sphere spectrum).