The common valuation conjecture for superrosy fields

Let $K$ be a field such that for every finite extension $L$ of $K$ and every natural number $n>0$, the index $[L^*:(L^*)^n]$ is finite. If $\operatorname{char}(K)=p>0$ and $f\colon L\to L$ is given by $f(x)=x^p-x$, also require that $[L^+:f[L]]$ is finite. The common valuation conjecture. Either $K$ admits a non-trivial definable valuation, or every non-trivial valuation on $K$ has divisible value group and algebraically or real closed residue field. The source describes this as a common approximation to several preceding conjectures; it remains unresolved in the supplied text.