Fleming-Viot particle systems describe populations of particles moving on a state space, where each particle that is killed is immediately replaced by a clone of a uniformly chosen surviving particle. The ancestral circulation refers to the measure tracking the long-run flow of lineage through the network, recording which nodes particles tend to pass through before being absorbed. The open problem is whether, for systems with more than two particles on networks of arbitrary topology, this ancestral circulation converges to a well-defined limit as the number of particles grows large, and whether that limit can be characterized explicitly in terms of the underlying killed Markov process.