Finite-state structure conjecture for 132-avoiding permutations with bounded adjacency

Let $S_n$ be the symmetric group, and let $A_n^{(m)}$ be the class

$$A_n^{(m)} = \{\pi \in S_n: \pi \text{ avoids } 132 \text{ and } |\pi_{i+1}-\pi_i| \le m\}.$$

Here, $m$ is a fixed integer with $m\geq 1$. Finite-state structure conjecture. The class $A_n^{(m)}$ admits a finite structural decomposition into a bounded collection of states determined by the local configuration near the beginning of the permutation. Transitions between these states depend only on this local information.

If true, the class would admit a finite transition-system description, implying a linear recurrence with constant coefficients for $\{A_n^{(m)}\}$ and a rational generating function. The conjecture is motivated by the complete structural analysis for $m=2$, but no general proof is supplied.