Characterization conjecture for universal classes of unlabeled robot systems

Let $G_i$ be unlabeled networks, let $k\geq 3$, and let $\mathcal F(G_i,k)$ denote the corresponding system of $k$ oblivious mobile robots. Let $G_i^\ast$ be the quotient graph associated with $G_i$, and call a set of systems universal when it can simulate every finite computation in the model considered. Characterization conjecture. The set

$$\left\{\mathcal F(G_i,k)\mid i\geq 0\right\}$$

is universal if and only if either the quotient graphs $G_i^\ast$ have unboundedly long sub-paths, or the graphs $G_i$ have unboundedly long shortest cycles.

This conjecture proposes that the two structural conditions identified by the preceding universality results exactly characterize universal classes of systems with at least three robots on unlabeled networks. The supplied text gives no resolution status.