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