Characterize completely regular spaces T with Esakia prime spectrum Spec(C(T))

Determine necessary and sufficient conditions on a completely regular topological space T under which the prime spectrum Spec(C(T)) of the ring of real-valued continuous functions on T is an Esakia space.

Background

The paper fully characterizes pseudocomplementation and Stone conditions for lattices attached to Spec C(T), and provides several equivalences for inverse spectra and z-spectra. However, the Esakia property of the spectral space Spec C(T) itself remains unresolved in general. For metric spaces, Corollary 5.9 shows that Spec C(T) is Esakia if and only if T is discrete, but a general characterization encompassing broader classes of completely regular spaces is not known.

The authors also note the lack of any known example of an infinite compact Hausdorff space T for which Spec C(T) is Esakia, underscoring the gap in understanding beyond the metrizable/discrete case.