Let $k$ be a field, and denote by $K_p(k)^{(n)}$ the weight $n$ eigenspace of the Adams operations on the $p$-th $K$-group of $k$.

The Beilinson-Soulé (BS) vanishing conjecture predicts that $$ K_{2q-p}(k)^{(q)}=0 $$ for $p \leq 0$ and $q>0$ (cf. Levine, "Tate motives and vanishing conjectures...").

If this conjecture holds for the field $k$ then (as demonstrated by Levine in ibid.) there exists a rigid abelian tensor category $MTM(k)$ of mixed Tate motives over $k$.

For $k$ a number field, the conjecture is true by results of Borel. Moreover, it is stated in Deligne-Goncharov's "Groupes Fondamentaux motiviques de Tate mixtes" at the beginning of section 1.6. that

"The only fields in characteristic zero for which the Beilinson-Soulé vanishing conjectures are proved are number fields, function fields of genus 0 curves over number fields and inductive limits of such fields."

**My Question:** What about fields of positive characteristic? Are there any for which BS vanishing is known (maybe the first thing to look at would be finite extensions of $\Bbb F_q(t)$ in keeping with the usual analogy with number fields)?

If not, are there at least partial (possibly negative) results in this direction? Or does the matter seem altogether hopeless?

Also, it would be nice to obtain a reference for the above claim in Deligne-Goncharov (but probably that's better off as a separate question).