Let $X$ be a compact (Hausdorff reduced) complex space. It is asserted (and used in an essential way) in a famous paper by Demailly and Păun ("Numerical characterization of the Kähler cone of a compact Kähler manifold") Proposition 3.3(iii), that a Kähler current $T$ on $X$ can be regularised by currents with analytic singularities. This result in turn is used in the proof of Theorem 0.1.

This seems rather surprising to me. In some special cases, for example when $T$ represents a Kähler class and $X$ is normal, this follows simply from a resolution argument. In general, is there a reference for this?