Jones' uniqueness conjecture for slowly oscillating periodic solutions of Wright's equation

Let

$$y'(t)=-\alpha y(t-1)\{1+y(t)\},$$

with parameter $\alpha>\tfrac{\pi}{2}$. A slowly oscillating periodic solution (SOPS) is a periodic solution $y(t)$ for which, up to time translation, there exist $q,\bar q>1$ and $L=q+\bar q$ such that $y(t)>0$ on $(0,q)$, $y(t)<0$ on $(q,L)$, and $y(t+L)=y(t)$ for all $t$, with $L$ the minimal period. Jones' conjecture. For every $\alpha>\tfrac{\pi}{2}$, Wright's equation has a unique SOPS up to time translation.

The existence of such solutions for all $\alpha>\tfrac{\pi}{2}$ was proved by Jones; this paper proves uniqueness for $\alpha\in[1.9,6.0]$, up to time translation, as a step toward the conjecture.