Projective Schur groups of Henselian fields

One of the open questions that has emerged in the study of the projective Schur group image of a field F is whether or not image is an algebraic relative Brauer group over F, i.e. does there exist an algebraic extension L/F such that image? We show that the same question for the Schur group of a number field has a negative answer. For the projective Schur group, no counterexample is known. In this paper we prove that image is an algebraic relative Brauer group for all Henselian valued fields F of equal characteristic whose residue field is a local or global field. For this, we first show how image is determined by image for an equicharacteristic Henselian field with arbitrary residue field k.