Orevkov's conjecture on empty outer ovals of M-curves with deep nests

Let $A$ be an $M$-curve with a deep nest, meaning a nest with at least three levels of nesting, of arbitrary degree. Orevkov's conjecture. $A$ must have some empty outer ovals. This generalizes the degree-nine restriction proved in the paper and predicts that every M-curve with a deep nest has empty ovals in the outer layer.