Negative-binomial conjecture for the number of geodesic paths

Let $G(n,\pi)$ be the unit disk graph, and let two vertices be displaced by $r_{xy}$. Write $\sigma_{r_{xy}}$ for the number of geodesic paths between them. Negative-binomial conjecture. The random variable $\sigma_{r_{xy}}$ is distributed as

$$\mathbb{P}(\sigma_{r_{xy}}=k)={k+r-1\choose k}p^{r}(1-p)^{k},$$

where $p$ and $r$ solve

$$\mathbb{E}(\sigma_{r_{xy}})=\frac{(1-p)r}{p},$$ $$\mathbb{E}(\sigma_{r_{xy}}^{2})=\mathbb{E}(\sigma_{r_{xy}})\left(\frac{1}{p}+\mathbb{E}(\sigma_{r_{xy}})\right).$$

This conjecture proposes a negative-binomial model for geodesic-path counts at larger displacements, where the variance exceeds the mean; the supplied text gives no resolution status.