Do bosons condense in a homogeneous magnetic field?

It has been known since the paper [26] and then due to a rigorous result [3] that the answer to the question in the title is negative for a three-dimensional "ideal gas of charged bosons". The present paper adds a new rigorous result in this direction. We show that the answer to the question becomes positive, if this "ideal gas of charged bosons" is simultaneously embedded in an appropriate periodic external potential. We prove that it is true for the Perfect Bose Gas (PBG), as well as for the Imperfect Bose Gas with a Mean-Field repulsive particle interaction.