Kołodziej-Tosatti's conjecture on compact Hermitian manifold with bounded mass property

In this note, we show a conjecture of Kołodziej-Tosatti about Morse-type integrals in nef $(1,1)$ classes on compact Hermitian manifold with bounded mass property. As a consequence, we give positive answers to Demailly-Păun’s conjecture and Tosatti-Weinkove’s conjecture when compact Hermitian manifold with bounded mass property.