Serre's conjecture II for principal homogeneous spaces

Let $K$ be a field of cohomological dimension at most $2$, and let $Z$ be a principal homogeneous space under a semisimple simply connected $K$-group. Serre's conjecture II. The variety $Z$ has a rational point. This remains open in general, although it is known for many classes of groups and fields, including several classical and exceptional types, $p$-adic fields, totally imaginary number fields, and various geometric fields.