Robber strategies and non-principal single ultrafilters

Let $G$ be a graph or digraph inducing a connectivity system, and consider an elimination game with an invisible robber in which cop positions satisfy the relevant connectivity bound. Robber-ultrafilter correspondence conjecture. For suitable graph-induced connectivity systems, robber-winning strategies in elimination games are closely related to non-principal single ultrafilters. The supplied text presents this as a conjectural relationship without specifying the precise hypotheses or equivalence, so its formulation and consequences remain open.