Monotonic growth-constant conjecture for 132-avoiding permutations with bounded adjacency

Let $A_n^{(m)}$ be the class of permutations in $S_n$ that avoid $132$ and satisfy $|\pi_{i+1}-\pi_i|\leq m$, with $m$ fixed. Define the exponential growth constant, when the limit exists, by

$$\alpha_m = \lim_{n\to\infty} (A_n^{(m)})^{1/n}.$$

Monotonic growth-constant conjecture. For fixed $m$, the limit $\alpha_m$ exists, and

$$1=\alpha_1<\alpha_2<\alpha_3<\cdots<4.$$

Moreover,

$$\lim_{m\to\infty}\alpha_m=4.$$

The known cases include growth rate $1$ for $m=1$, growth constant approximately $1.46557$ for $m=2$, and the Catalan growth rate $4$ when the adjacency bound is at least $n-1$. The strict monotonicity and limiting assertion remain conjectural in the source.