Question : Do there exist infinitely many rational numbers $0\lt p\lt 1$ which are simply normal to two co-prime bases? How about three or more co-prime bases?
Remark : This question has been asked previously on math.SE without receiving any answers.
Motivation & Examples : At math.SE, I knew that the numbers I had been interested in are called the numbers which are simply normal to co-prime bases. My first question at math.SE 'Does there exist a rational number $0\lt p\lt 1$ which is simply normal to two or three co-prime bases?' has already been solved because the following examples have been found. The answer for the question at the top, which is my second question at math.SE, seems yes, but I don't know how to prove it. I would like to know how to prove or disprove it and any relevant references. $$\frac{5}{26}=0.0\overline{011000100111}_2=0.\overline{012}_3$$ $$\frac{1}{66}=0.0\overline{0000011111}_2=0.\overline{0014213324}_5$$ $$\begin{align}\frac{19}{679}&=0.\overline{000202101202221210210100112200011122121121210002}_{3}\\&=0.\overline{001302213121202301331023}_{4}\end{align}$$ $$\begin{align}\frac{9}{142}&=0.\overline{01243}_5\\&=0.\overline{0305114246565163130022623412213461063615524201015035366440432544532056}_7\end{align}$$ $$\begin{align}\frac{809}{2046}&=0.0\overline{1100101001}_2\\&=0.1\overline{012000202022100112121102121012}_3\\&=0.\overline{144203101322123012334420003441}_5\end{align}$$
By the way, finding a rational number $0\lt p\lt 1$ which is simply normal to both base $4$ and base $5$ may not be easy. (Another interesting question would be 'Does there exist a rational number $0\lt p\lt 1$ which is simply normal to any two co-prime bases?') Also, I would like to know if there is a rational number which is simply normal to four co-prime bases.