Constant Chern holomorphic sectional curvature on two-step solvmanifolds

Let $M=Γ\backslash G$ be a compact solvmanifold with a left-invariant Hermitian structure whose Lie algebra is two-step solvable. If the Chern holomorphic sectional curvature is constant, then the derived algebra is $J$-invariant and the Chern connection is flat. At the Lie-algebra level, we also prove that constant Chern holomorphic sectional curvature is impossible for a unimodular two-step solvable Hermitian Lie algebra of mixed Freibert-Swann type.