  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.
