On rationally connected varieties over $C_1$ fields of characteristic 0

In the 1950s Lang studied the properties of $C_1$ fields, that is, fields over which every hypersurface of degree at most n in a projective space of dimension n has a rational point. Later he conjectured that every smooth proper rationally connected variety over a $C_1$ field has a rational point. The conjecture is proven for finite fields (Esnault) and function fields of curves over algebraically closed fields (GraberHarrisde JongStarr), but it is still open for the maximal unramified extensions of $p$-adic fields. I use birational geometry in characteristic 0 to reduce the conjecture to the problem of finding rational points on Fano varieties with terminal singularities, and I provide some evidence in dimension 3.