Covers of varieties with completely split points

LetK be a field such that all Sylow subgroups of its absolute Galois groupG _Kare infinite. LetX be a smooth variety overK with function fieldF andY→X the normalisation in a finite, separable extensionE/vbF. We show: If there is a closed pointx∈X which does not split completely inY→X, then the set of these points is Zariski dense inX.