that for $\alpha > {\tfrac{\pi}{2}}$ there exists at least one *slowly oscillating periodic solution (SOPS)*. That is, a periodic solution $y:{\mathbb{R}}\to {\mathbb{R}}$ which is positive for at least one unit of time (the delay time in Wright’s equation), negative for at least one unit of time, and then repeats. In this paper we prove there is a unique SOPS to for $ \alpha \in ( {\tfrac{\pi}{2}},1.9]$, thus completing a proof of Jones’ conjecture:

\[prop:Jones\] For every $ \alpha > {\tfrac{\pi}{2}}$ there exists a unique slowly oscillating periodic solution to .

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This work contributes a capstone to many decades of mathematical work studying Wright’s equation. To briefly review, a *principal branch* of slowly oscillating periodic orbits is born at $ \alpha = {\tfrac{\pi}{2}}$ and continues on for all $ \alpha > {\tfrac{\pi}{2}}$ [@nussbaum1975global]. Moreover, Wright’s equation has supercritical Hopf bifurcations at $ \alpha = {\tfrac{\pi}{2}}+ 2 n\pi$ for integers $ n \geq 0$, with slowly oscillating periodic orbits arising when $n=0$, and rapidly oscillating periodic orbits arising when $n\geq1$ (see Figure \[fig:BifurcationDiagram\]) [@chow1977integral]. Together with the parameter $\alpha$, the collection of periodic orbits forms a 2-dimensional manifold [@regala1989periodic].

A two-part geometric version of Jones’ conjecture was proposed in [@lessard2010recent]: (i) 