  Let $(X,\omega)$ be an $n$-dimensional compact Hermitian manifold with
$\omega$ a pluriclosed Hermitian metric, i.e. $dd^c\omega=0$. Let
$\{\alpha\},\{\beta \}\in H^{1,1}_{BC}(X,\mathbb R)$ be two nef classes, such
that $\alpha^n-n\alpha^{n-1}\cdot\beta>0$. In this short note, we prove that if
there is a bounded quasi-plurisubharmonic potential $\rho$, such that
$\alpha+dd^c\rho\geq 0$ in the weak sense of currents, then the class
$\{\alpha-\beta\}$ contains a K\"ahler current. This gives a partial solution
of Demailly's transcendental Morse inequalities conjecture.
