The bifurcation-locus conjecture for the periodgon

Let $\dot z=P_{\epsilon}(z)$ be the vector-field family under consideration, with reduced parameters $(s,\theta,\alpha)$ on the real $3$-sphere corresponding to $\|\epsilon\|=1$. The bifurcation locus of the periodgon is the set

$$\Sigma=\{(s,\theta,\alpha): s\in[0,\tfrac12],\ \theta\in\tfrac{\pi}{k}\mathbb{Z}\}.$$

Bifurcation-locus conjecture. The bifurcation locus of the periodgon of this family is precisely $\Sigma$.

This conjecture identifies the parameters where the shape of the star domain, and hence of the periodgon, changes discontinuously. The preceding discussion explains that bifurcations arise from changes in the ordering of periodgon sides or from multiple homoclinic loops; the conjecture specifies the corresponding locus in the reduced parameter space.