Strict positivity of K\"ahler-Einstein currents

K\"ahler-Einstein currents, also known as singular K\"ahler-Einstein metrics, have been introduced and constructed a little over a decade ago. These currents live on mildly singular compact K\"ahler spaces $X$ and their two defining properties are the following: they are genuine K\"ahler-Einstein metrics on $X_{\rm reg}$ and they admit local bounded potentials near the singularities of $X$. In this note we show that these currents dominate a K\"ahler form near the singular locus, when either $X$ admits a global smoothing, or when $X$ has isolated smoothable singularities. Our results apply to klt pairs and allow us to show that if $X$ is any compact K\"ahler space of dimension $3$ with log terminal singularities, then any singular K\"ahler-Einstein metric of non-positive curvature dominates a K\"ahler form.