  Transcendental holomorphic Morse inequalities aim at characterizing the
positivity of transcendental cohomology classes of type $(1,1)$. In this paper,
we prove a weak version of Demailly's conjecture on transcendental Morse
inequalities on compact K\"ahler manifolds. And as a consequence, we partially
improve a result of Boucksom-Demailly-Paun-Peternell.
