Caracol graph Ehrhart polynomial product conjecture

Let $\textup{Car}_{n+1}$ be the Caracol graph on $n+1$ vertices, and let $E_{\textup{Car}_{n+1}}(x)$ denote the Ehrhart polynomial associated with its flow polytope. Caracol Ehrhart product conjecture. For the Caracol graph $\textup{Car}_{n+1}$ we have

$$E_{\textup{Car}_{n+1}}(x)=\frac{1}{xn+n-3}\binom{xn+2n-5}{n-1}\binom{x+n-3}{n-2}.$$

This is presented as a conjectured product formula; the source gives no resolution of it.