A note on Demailly's transcendental Morse inequalities conjecture

Let $(X,\omega )$ be an $n$-dimensional compact Hermitian manifold with $\omega$ a pluriclosed Hermitian metric, i.e. $d{d}^c\omega =0$. Let $\{\alpha \},\{\beta \}\in H_{BC}^{1,1}(X,\mathbb{R})$ 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 $\alpha +d{d}^c\rho \ge 0$ in the weak sense of currents, then the class $\{\alpha -\beta \}$ contains a Kähler current. This gives a partial solution of Demailly’s transcendental Morse inequalities conjecture.