Small Digitwise perturbations of a number make it normal to unrelated bases

Let $r,g\ge 2$ be integers such that $\log g/\log r$ is irrational. We show that under $r$-digitwise random perturbations of an expanded to base $r$ real number $x$, which are small enough to preserve $r$-digit asymptotic frequency spectrum of $x$, the $g$-adic digits of $x$ tend to have the most chaotic behaviour.