Normality conjecture for self-similar measures

Let $\{\varphi_i(x)=\lambda_i x+t_i\}_{i\in \mathcal{I}}$ be an iterated function system of contracting similarities, and let $b\geq 2$ be an integer. Suppose that

$$\frac{\log b}{\log |\lambda_i|}\notin\mathbb{Q}$$

for some $i\in\mathcal{I}$. A point $x$ is normal in base $b$ if every finite block of base-$b$ digits occurs in its expansion with the expected limiting frequency. Normality conjecture. Almost every $x$ is normal in base $b$ with respect to any non-atomic fully supported self-similar measure of $\{\varphi_i\}_{i\in\mathcal{I}}$. The conjecture is false without the stated irrationality assumption on the contraction ratio; thus the claim as stated is refuted, although the version under the irrationality hypothesis is the asserted result of interest.