The proposed growth-control conjecture for motivic cohomology of fields

The paper considers motivic cohomology groups $\operatorname H_{\mathrm M}^{n,i}(K)$ of a field $K$, with $\delta(K)$ denoting its Kronecker dimension. Growth-control conjecture. For every field $K$ finitely generated over its prime field $F$, one has

$$\operatorname H_{\mathrm M}^{n,i}(K)=0$$

unless $(n,i)=(0,0)$ or $n\in[1,\delta(K)]$. This conjecture proposes a precise vanishing range complementing the Beilinson–Soulé conjecture, in a setting where motivic cohomology can otherwise have infinite rank.