In 1986 Kirchberg showed that each eigenvalue of the Dirac operator on a compact Kähler manifold of even complex dimension satisfies some inequality involving the scalar curvature. It is conjectured that the manifolds for the limiting case of this inequality are products T^2×N, where T^2 is a flat torus and N is the twistor space of a quaternionic Kähler manifold of positive scalar curvature. In 1990 Lichnerowicz announced an affirmative answer for this conjecture, but his proof seems to work only when assuming that the Ricci tensor is parallel. The aim of this note is to prove several results about manifolds satisfying the limiting case of Kirchberg''s inequality and to prove the above conjecture in some particular cases.