The homoclinic-orbit simplicity conjecture

Let $G$ be the group under consideration, and call a real-analytic subset of $G$ simple if it is contained in a finite union of sets of the form

$$\mathcal S(\chi-c)=\{g\in G:\chi(g)-c=0\},$$

with $\chi\in\widehat G$ and $c\in\mathbb T_c$. A homoclinic orbit is an orbit of the homoclinic action referred to in the paper. The homoclinic-orbit simplicity conjecture. Every real-analytic subset of $G$ that intersects every homoclinic orbit is simple. This assertion is presented as an assertion whose validity would prove the paper's preceding conjecture; no resolution is given here.