Vaisman's theorem and local reducibility

As proven in a celebrated theorem due to Vaisman, pure locally conformally K\"ahler metrics do not exist on compact K\"ahler manifolds. In a previous paper, we extended this result to the singular setting, more precisely to K\"ahler spaces which are locally irreducible. Without the additional assumption of local irreducibility, there are counterexamples for which Vaisman's theorem does not hold. In this article, we give a much broader sufficient condition under which Vaisman's theorem still holds for compact K\"ahler spaces which are locally reducible.