Continuous approximation of quasi-plurisubharmonic functions

Let $X$ be a compact K\"ahler manifold and $\theta$ a smooth closed $(1,1)$-real form representing a big cohomology class $\alpha \in H^{1,1}(X,\R)$. The purpose of this note is to show, using pluripotential and viscosity techniques, that any $\theta$-plurisubharmonic function $\f$ can be approximated from above by a decreasing sequence of continuous $\theta$-plurisubharmonic functions with minimal singularities, assuming that there exists a single such function.