Reformulated Wright conjecture on the principal branch and isolas of slowly oscillating periodic solutions

Consider Wright's equation

$$y'(t)=-\alpha y(t-1)[1+y(t)],\qquad \alpha>0.$$

Let $\mathcal{F}_0$ be the branch of slowly oscillating periodic solutions (SOPS) that bifurcates forward in $\alpha$ at $\alpha=\frac{\pi}{2}$. A fold is a turning point of this branch in the parameter $\alpha$, and an isola is a connected component of the set of SOPS separate from $\mathcal{F}_0$. Reformulated Wright conjecture. The branch $\mathcal{F}_0$ has no fold for $\alpha\in(\frac{\pi}{2},5.67]$, and there are no isolas of SOPS for $\alpha\in(\frac{\pi}{2},5.67]$. This reformulation isolates the remaining part of the uniqueness question after uniqueness was established for $\alpha>5.67$. The branch $\mathcal{F}_0$ is known to be a regular curve because it has no secondary bifurcation, and the paper reports a validated-continuation result proving that part of the branch has no fold; the complete assertions remain open.