Buchstaber–Tertychnyi garland conjecture for phase-lock areas

For every $r\in\mathbb Z$, let $L_r$ denote the phase-lock area with rotation number $r$, and set

$$L_r^+=L_r\cap\{A\geq 0\}.$$

Let $\mathcal A_{r,1},\mathcal A_{r,2},\ldots$ denote the adjacencies in the line $\{B=r\omega\}$, ordered by their $A$-coordinates. Buchstaber–Tertychnyi garland conjecture. The upper part $L_r^+$ of each phase-lock area $L_r$ is a garland of infinitely many connected components separated by these adjacencies. The conjecture is motivated by numerical simulations and prior theoretical results; the source gives no resolution evidence for it.