  We describe a method to show that certain elliptic surfaces do not admit
purely inseparable multisections (equivalently, that genus one curves over
function fields admit no points over the perfect closure of the base field) and
use it to show that any non-Jacobian elliptic structure on a very general
supersingular K3 surface has no purely inseparable multisections. We also
describe specific examples of such fibrations without purely inseparable
multisections. Finally, we discuss the consequences for the claimed proof of
the Artin conjecture on unirationality of supersingular K3 surfaces.
