The attracting-periodic-orbit conjecture for equilibrium types

Consider the piecewise-defined system of equations and classify its attracting equilibria as regular, boundary, or virtual according to their locations relative to the switching domains. Attracting-periodic-orbit conjecture. If the system has either two attracting boundary equilibria, two attracting virtual equilibria, or one boundary and one virtual attracting equilibrium, then it has an attracting periodic orbit. Conversely, if one of the attracting equilibria is regular, then the system has no attracting periodic orbit. The supplied text gives a proof idea and numerical support, but no proof or resolution.