Kato–Kuzumaki's cohomological dimension conjecture

Let $k$ be a field. For non-negative integers $i$ and $q$, the property $C_i^q$ means that for every pair of integers $d,n\geq 1$, every finite extension $k'/k$, and every hypersurface $Z\subseteq\mathbf{P}^n_{k'}$ of degree $d$ with $d^i\leq n$, one has

$$\mathrm{K}_q(k')=\mathrm{N}_q(Z/k').$$

Here $\mathrm{N}_q(Z/k')$ is the subgroup of $\mathrm{K}_q(k')$ generated by the norm images from finite extensions $l/k'$ for which $Z(l)\neq\emptyset$.

Kato–Kuzumaki's conjecture. A field $k$ satisfies $C_i^q$ if and only if

$$\operatorname{dim} k\leq i+q.$$

The conjecture was proposed as a Diophantine characterization of the cohomological dimension by means of Milnor $\mathrm{K}$-theory. It is false in full generality, with counterexamples due to Merkurjev and to Colliot-Thélène and Madore, although it remains open for arithmetically interesting fields.