Existence of smaller automata when restricted to inputs q = b^i

Determine whether there exist deterministic finite automata with output having fewer states than the Walnut-constructed DFA that accepts the Zeckendorf representation of q and outputs Δ(q) = ⌊b q φ⌋ − b ⌊q φ⌋, which remain correct on the restricted input set q = b^i for i ≥ 0 (equivalently, on inputs corresponding to powers of b used to compute the n-th base-b digit of φ), even if such automata differ on other inputs.

Background

Walnut constructs a DFA that, on input the Zeckendorf representation of q, accepts precisely when the pair (q, ⌊q φ⌋) satisfies the synchronized relation, and this yields a DFA for Δ(q) = ⌊b q φ⌋ − b ⌊q φ⌋ that is minimal for all q ≥ 0. However, the digit-computing application only uses inputs q of the special form b^i.

The authors ask whether restricting to the powers of b admits strictly smaller automata than the globally minimal one. They note the problem is likely difficult (being a special case of minimal DFA identification from incomplete data, which is NP-hard) and use SAT-based methods to prove minimality in some instances, but state that the general answer is unknown.