A non-geodesic edge that survives in the loop-erased ant process

In the loop-erased ant process of Kious, Mailler and Schapira, ants walk one after another from a nest $N$ to a food source $F$ on a finite graph, each ant stepping along an edge with probability proportional to its weight and then adding one to the weight of every edge of the loop-erasure of its time-reversed trajectory. It was conjectured that the weight of an edge grows linearly exactly when the edge lies on a geodesic from $N$ to $F$. Let $G_\ell$ be the graph made of a triangle $N,u,v$ and two paths of $\ell$ edges, one from $u$ to $F$ and one from $v$ to $F$. We prove that on $G_\ell$ the normalised weights converge almost surely. The limit is $\rho$ on the edges of the path $N,u,\dots,F$, it is $1-\rho$ on the edges of the path $N,v,\dots,F$, and it is $(1-1/\ell)\rho(1-\rho)$ on the edge $uv$, where $\rho$ is a non-degenerate random variable with values in $(0,1)$. For $\ell\ge2$ the edge $uv$ lies on no geodesic and keeps a positive share of the weight, so the conjecture fails on $G_\ell$, and the proportion of ants that reinforce a path which is not a geodesic has a positive limit. For $\ell=1$ the graph is the losange of Kious, Mailler and Schapira, the edge $uv$ dies, and the theorem gives for the loop-erased rule the convergence that they proved for the uniform-geodesic rule. The forward-loop-erased ant process of Armfield and Mailler, in which the ant reinforces the loop-erasure of its trajectory instead of that of the reversed trajectory, has the same law, so all of these statements hold for it as well.