Interpretations of Büchi arithmetics in themselves

Büchi arithmetic BA$_k,\ k \geq 2,$ is the elementary theory of the natural numbers with equality, addition and the function $V_k (x)$ which returns the largest power of 2 dividing $x$. These arithmetics represent finite automata in the following sense: sets of natural numbers definable in BA$_k$ are exactly those accepted by some finite automaton when expressed in $k$-ary digits. We study interpretations of Büchi arithmetics in themselves. For Presburger arithmetic (theory of natural numbers with addition), it was previously proven by the author and F. Pakhomov that every self-interpretation is definably isomorphic to the identical interpretation.