Algebraic independence conjecture for automatic numbers in independent bases

Let $r\geq 1$ be an integer. Let $b_1,\ldots,b_r$ be multiplicatively independent positive integers, and, for every $i$, $1\leq i\leq r$, let $\xi_i$ be a real number that is automatic in base $b_i$. Algebraic independence conjecture. Unless one of the $\xi_i$ is rational, the numbers $\xi_1,\ldots,\xi_r$ are algebraically independent over $\overline{\mathbb Q}$. The case $r=1$ was proved by Bugeaud and the first author using the subspace theorem; the general multivariable statement is the proposed strengthening.