Define the 'reinterpret $x$ from base $a$ to base $b$' function $R_{a,b}(x)$ as $$ R_{a,b}(x)=\sum_{i\in\mathbb{Z}}b^iD_{a,i}(x) $$ where $D_{m,i}(x)$ is the $i$-th term in the $m$-ary expansion of $x$, $$ D_{m,i}(x) = \left\lfloor m(m^{i-1}x - \lfloor m^{i-1}x\rfloor) \right\rfloor. $$
For integers $a,b>1$, is it true that $x$ is transcendental iff $R_{a,b}(x)$ is transcendental?
For example, is 3.050330051415124105234414053125321..., $\pi$ reinterpreted from base 6 to base 10, transcendental?
Jyrki Lahtonen suggests a useful reformulation: write the base-$a$ digits of a number $x$ as a Laurent polynomial $f$ such that $f(1/a)=x$. Then the question is whether $f(1/a)$ is transcendental if and only if $f(1/b)$ is transcendental, for integers $a,b>1.$
This is probably asking too much; my hope was to prove that, say, $R_{a,b}(\pi)$ is transcendental for all integers $a,b>1$ (or substitute your favorite transcendental number). Any movement in this direction would be appreciated.