In Demailly's Complex Analytic and Differential Geometry, prop.4.1 Chapter III, he states that
a)$T$ has non zero bidimension (p,p) (i.e. degree of $T < 2n$, where $T$ is a closed positive current.)
b)$X$ is covered by a family of Stein open sets $\Omega \subset \subset X$ whose boundaries $\partial \Omega$ do not intersect $L(u) \cap \mathrm{Supp} T$
Then the current $uT$ has locally finite mass in X
At the beginning of the proof, he asserts that by shrinking $\Omega$ slightly, we may assume that $\Omega$ has a smooth strongly pseudoconvex boundary. But how can the shrinking be achieved?
You can shrink $\Omega$ to be small enough so that it looks like a ball in $\mathbb{C}^n$. Such a thing has a smooth strongly pseudoconvex boundary.