Separation conjecture for adjacent spine vertices

Let $G$ be an augmented caterpillar drawn as a great-circle thrackle, and let $v$ and $v'$ be adjacent vertices in the spine of $G$. A separating $2$-path at a vertex is a $2$-path that separates at that vertex. Separation conjecture for adjacent spine vertices. If some $2$-path separates at $v$, then no $2$-path separates at $v'$. The conjecture further specifies the possible structure of great-circle thrackleable augmented caterpillars; the source leaves it open and suggests that crossing orientations could help prove or refute it.