Continuity conjecture for similarities of the almost Mathieu rational butterfly

Let the full butterfly be the parameterized spectrum of the almost Mathieu operators, including irrational values of the vertical parameter $\theta$, and let $S=S_{M,r,\pm}$ be a similarity of the rational butterfly as described in Section 10. Continuity conjecture. There is a unique continuous extension of $S$ to the full butterfly, possibly double-valued at certain points along the vertical line $x=0$. This conjecture proposes that the similarity structure established for the rational butterfly persists by continuity on the full butterfly; the only possible nonuniqueness is at specified points on the line $x=0$.