Automaticity conjecture for the sequence of integer parts of consecutive-term ratios

Let $(h_n)_{n\in\mathbb{N}}$ be the sequence studied in the paper, and define

$$a(n)=\left\lfloor\frac{|h_n|}{|h_{n-1}|}\right\rfloor.$$

Automaticity conjecture. The sequence $(a(n))_{n\in\mathbb{N}}$ is $2$-automatic. The paper establishes that $a(n)\in\{0,1,2,3,4\}$, while the stronger automaticity assertion remains open.