The Fleming-Viot particle system is a finite population model where particles diffuse on a domain and, upon hitting the boundary (being killed), are instantly reborn at the location of a uniformly chosen surviving particle. For two particles, researchers have recently derived explicit formulas describing the long-run ancestral circulation, meaning the statistical pattern of which particle line of descent dominates over time. The open problem is whether analogous closed-form or tractable characterizations of the stationary ancestral circulation exist for systems of three or more particles, where the combinatorial and analytic complexity explodes dramatically due to simultaneous interaction among multiple lineages.