The Hasson–Shelah definable valuation conjecture for NIP fields

Let $K$ be an infinite field with NIP such that, for every finite extension $L$ of $K$ and every natural number $n>0$, the index $[L^*:(L^*)^n]$ is finite. The Hasson–Shelah definable valuation conjecture. Either $K$ admits a non-trivial definable valuation, or $K$ is algebraically closed or real closed. The conjecture concerns a dichotomy between definable valuations and strong algebraic restrictions on NIP fields; its status is not resolved in the supplied text.