Kahn's conjecture on Somekawa K-groups of semi-abelian varieties

Let $k$ be a field finitely generated over its prime subfield $k_0$. If $k$ has characteristic zero, set

$$s=\operatorname{trdeg}(k/k_0)+1;$$

if $k$ has positive characteristic, set

$$s=\operatorname{trdeg}(k/k_0).$$

Let $G_1,\ldots,G_r$ be semi-abelian varieties over $k$.

Kahn's conjecture. If $r=s+1$, then the Somekawa K-group $K(k;G_1,\ldots,G_r)$ is torsion. If $r>s+1$ and $k$ does not admit an ordered field structure, then

$$K(k;G_1,\ldots,G_r)=0.$$

This predicts a sharp relationship between the transcendence degree of a finitely generated field and the torsion or vanishing of Somekawa K-groups. The supplied text gives no resolution status for the conjecture.