Disjoint finite geodesics in first-passage percolation

We investigate first-passage percolation on the lattice ${\mathbb{Z}}^d$ for dimensions $d\ge 2$. Each edge $e$ of the graph is assigned an independent copy of a non-negative random variable $\tau$. We only assume $\mathrm{\P }[\tau =0]0$ is explicit) for the probability of having two disjoint geodesics between two pairs of neighbouring vertices at distance $n$. Additionally, under more specific assumptions on the distribution of $\tau$, we obtain similar lower bounds for the probability of having two disjoint geodesics (except for their starting and ending points) between the same two vertices.