Vaisman solvmanifolds as finite quotients of Kodaira-Thurston nilmanifolds

We prove that every Vaisman solvmanifold is a finite quotient of a Kodaira-Thurston manifold. More generally, we show that every aspherical compact Vaisman manifold with strongly polycyclic fundamental group is a finite quotient of a Kodaira-Thurston manifold. As consequences, we obtain that every completely solvable solvmanifold admitting a Vaisman structure is a Kodaira-Thurston manifold, that Oeljeklaus-Toma manifolds admit no Vaisman structures (not necessarily left-invariant), and that solvmanifolds does not admit LCK Einstein-Weyl structures.