Finiteness of open gaps in the local Iwatsuka model

Consider the local Iwatsuka model with a nonzero magnetic field $b$ satisfying minimal regularity assumptions such as those formulated in Section 2.1. Finiteness conjecture. The number of open gaps in the local Iwatsuka model is finite. The preceding example establishes this conclusion under a particular sign and endpoint condition, while the general case is left for a deeper analysis of the behaviour around classical turning points.