Dichotomy conjecture for negative-Schwarzian delay equations

Assume that $a\geq 0$ and that $f$ is an $S$-map. Let $K$ be the unique fixed point of $f$. Equation (1) has an equilibrium at $K$; a global attractor attracts the phase space, and a stable periodic orbit is a periodic orbit that attracts nearby solutions. Dichotomy conjecture for negative-Schwarzian delay equations. Either: (a) the equilibrium $K$ is a global attractor whenever it is asymptotically stable; or (b) if $K$ is unstable, there exists a unique stable periodic orbit of Equation (1), and this orbit attracts an open and dense subset of the phase space. The conjecture seeks to extend the map-level dichotomy to the associated delay equation. The source presents it as motivated by numerical experiments and by the corresponding dichotomy theorem for $S$-maps; it remains open in the stated generality.