Asymptotic ratio conjecture for spanning trees of suitable graphs

Let $J_{n,m}$ be the graph whose number of spanning trees is denoted by $\sigma(J_{n,m})$, and let $n,m ein\mathbb{N}$ with fixed $n\geq 2$. Define

$$a_{n,m}=\frac{\sigma(J_{n,m+1})}{\sigma(J_{n,m})}.$$

Asymptotic ratio conjecture. The limit exists:

$$\lim_{m\to +\infty}a_{n,m}=\delta_n,$$

where $\delta_n\in\mathbb{R}$. Moreover,

$$\sigma(J_{n,m})=(\delta_n)^{m-3}\sigma(J_{n,3}).$$

The conjecture proposes a limiting multiplicative growth rate for the number of spanning trees as the second graph parameter tends to infinity. The preceding numerical values suggest this behavior for $n=2$ and $n=3$, but no proof or resolution is supplied in the source.