Conjecture on limiting pheromone weights and geodesics

Let $\mathcal G=(V,E)$ be any finite graph with two distinct marked vertices $N$ and $F$. For each edge $e\in E$, let $W_e(n)$ be its pheromone weight after $n$ ants, and let a geodesic mean a shortest path from $N$ to $F$ in $\mathcal G$. The loop-erased and uniform-geodesic ant processes are the two return rules described in the model.

Limiting-weight conjecture. Almost surely, as $n\to+\infty$, for every $e\in E$,

$$\frac{W_e(n)}{n}\to\chi_e,$$

where $(\chi_e)_{e\in E}$ is a random vector satisfying: for the loop-erased ant process, $\chi_e\ne0$ almost surely if and only if $e$ belongs to at least one geodesic from $N$ to $F$; for the uniform-geodesic ant process, $\chi_e\ne0$ almost surely only if $e$ belongs to at least one geodesic from $N$ to $F$. Consequently, if the geodesic $\gamma$ is unique, then almost surely $\chi_e=\boldsymbol 1_{e\in\gamma}$ for every $e\in E$ in both models.

This gives a precise formulation of geodesic selection through the limiting normalized pheromone weights. The source indicates proofs for particular graph classes, but does not state a resolution of the conjecture for arbitrary finite graphs.