Conjecture on smooth arithmetical structures on paths with a doubled edge

Let $\mathcal{P}_{m,n}$ be the graph consisting of a path with $m$ vertices on one side of a doubled edge and $n$ vertices on the other. A smooth arithmetical structure on $\mathcal{P}_{m,n}$ is an arithmetical structure whose vertex labels satisfy $a_1>a_2>\ldots>a_m$ and $b_1>b_2>\ldots>b_n$; write $\operatorname{SArith}(\mathcal{P}_{m,n})$ for the set of such structures.

Smooth-structure growth conjecture. The number of smooth arithmetical structures on $\mathcal{P}_{m,n}$ grows at the same rate as

$$\binom{m+n-1}{n}.$$

The paper establishes this polynomial growth rate in the cases $m=1,2,3$, and the conjecture proposes the corresponding binomial-coefficient asymptotic for general $m$; the authors leave its further investigation open.