Arithmetical structures conjecture on extremal graph orders

Let $G$ be a simple graph with $n$ vertices. For a graph $H$, let $\mathcal{A}(H)$ denote its set of arithmetical structures, and let $|\mathcal{A}(H)|$ be its cardinality. Let $P_n$ and $K_n$ be the path and complete graph, respectively, on $n$ vertices.

Arithmetical structures conjecture.

$$\big|\mathcal{A}(P_n)\big|\leq \big|\mathcal{A}(G)\big|\leq \big|\mathcal{A}(K_n)\big|.$$

This conjecture predicts that among simple graphs on $n$ vertices, the path minimizes and the complete graph maximizes the number of arithmetical structures. The paper presents computational evidence for the claim; no resolution is supplied here.