Bounds for non-isomorphic maximal Arrow single-peaked domains

Let $A$ be a set of size $m$. Let $\mathcal{N}(m)$ denote the number of non-isomorphic maximal Arrow single-peaked domains on $A$, and define $\mathcal{P}(m)$ by

$$\mathcal{P}(3)=1,\qquad \mathcal{P}(m)=2^{m-3}\mathcal{P}(m-1)\quad\text{for }m\geq 4.$$

The conjectured bounds. The number $\mathcal{N}(m)$ satisfies

$$\frac{1}{2}\bigl(\mathcal{P}(m)+\mathcal{P}(m-1)\bigr)\leq \mathcal{N}(m)\leq \frac{1}{2}\bigl(\mathcal{P}(m)+2\mathcal{P}(m-1)\bigr).$$

The conjecture concerns the enumeration of maximal Arrow single-peaked domains up to isomorphism; the preceding discussion motivates it by predicting that domains with inherited permutation $\sigma=id$ comprise at least half of all self-paired domains. Its resolution is not established in the supplied text.