The automaticity conjecture for generalised-polynomial sequences

Let $f\colon\mathbb{N}_0\to\Omega$ be a sequence, where $\Omega$ is a finite alphabet. Call $f$ automatic if it is generated by a finite automaton, and call it a generalised-polynomial sequence if its values are given by a generalised polynomial. A sequence is ultimately periodic if it agrees with a periodic sequence outside a finite set. Automaticity conjecture. If a sequence $f$ is simultaneously automatic and generalised polynomial, then $f$ is ultimately periodic. This is the sequence formulation of the paper's motivating conjecture. The authors note that the Fibonacci word supplies a nontrivial generalised-polynomial sequence for a Zeckendorf-type notion of automaticity, while the conjecture concerns the usual positional notion; partial results are proved, but the general statement remains open.