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We study several problems motivated by Crouzeix's conjecture, which weconsider in the special setting of model spaces and compressions of the shiftwith finite Blaschke products as symbols. We pose a version of the conjecturein this setting, called the level set Crouzeix (LSC) conjecture, and establishstructural and uniqueness properties for (open) level sets of finite Blaschkeproducts that allow us to prove the LSC conjecture in several cases. Inparticular, we use the geometry of the numerical range to prove the LSCconjecture for compressions of the shift corresponding to unicritical Blaschkeproducts of degree $4$.

Blaschke Products, Level Sets, and Crouzeix's Conjecture