Uniform distribution conjecture for powers under self-similar measures

Let $\mu$ be a non-atomic self-similar measure with support contained in $[1,\infty)$. A sequence $(y_n)_{n=1}^{\infty}$ is uniformly distributed modulo one if, for every interval $[a,b)\subseteq[0,1)$, the proportion of indices $n\leq N$ for which the fractional part of $y_n$ lies in $[a,b)$ tends to $b-a$ as $N\to\infty$. Self-similar equidistribution conjecture. For $\mu$ almost every $x$, the sequence $(x^n)_{n=1}^{\infty}$ is uniformly distributed modulo one. The conjecture proposes an analogue of Koksma's almost-everywhere equidistribution theorem for the natural class of self-similar measures, whose affine construction is viewed as independent of the nonlinear maps $x\mapsto x^n$.