On the extension of quasiplurisubharmonic functions

Let $(V,\omega)$ be a compact K\"ahler manifold such that $V$ admits a cover by Zariski-open Stein sets with the property that $\omega$ has a strictly plurisubharmonic exhaustive potential on each element of the cover. If $X\subset V$ is an analytic subvariety, we prove that any $\omega|_X$-plurisubharmonic function on $X$ extends to a $\omega$-plurisubharmonic function on $V$. This result generalizes a previous result of ours on the extension of singular metrics of ample line bundles. It allows one to show that any transcendental K\"ahler class in the real Neron-Severi space $NS_{\mathbb R}(V)$ has this extension property.