Esakia status of Spec(C(βN))

Ascertain whether the prime spectrum Spec(C(βN)) of the ring of real-valued continuous functions on the Stone–Čech compactification βN of the natural numbers is an Esakia space.

Background

Within the broader open problem of characterizing all T with Esakia Spec C(T), the authors single out the specific case of T = βN. They report no known example of an infinite compact Hausdorff T with Esakia Spec C(T), making βN a natural test case.

Resolving this question would provide insight into whether any infinite compact Hausdorff examples exist and would inform the general characterization problem.