Let $\psi$ be a positive singular metric of $L$ on $X$. Then $\psi-h$ is a globally well defined $\omega$-psh function. It is u.s.c. hence bounded from above on $X$: we let $C_{\psi}$ denotes its maximum. Then $\psi-h-C_{\psi}\leq 0$ on $X$, hence $\psi-h\leq V_{X,\omega}^{*}+C_{\psi}$, which yields $\psi\leq h_{min}+C_{\psi}$. ∎