The two-base automaticity conjecture

Let $p$ and $q$ be two multiplicatively independent natural numbers, and let automatic in base $b$ mean that the real number's expansion in base $b$ can be generated by a finite automaton. The two-base automaticity conjecture. A real number cannot be automatic in both bases $p$ and $q$, unless it is rational. This is a computational analogue of Furstenberg's conjecture: automaticity in a base implies zero entropy in that base. The claim remains challenging; the source notes that no real number had then been proved to be automatic in one base and not automatic in another.