Counting Boolean antichains with interval order filters

Count Boolean antichains whose associated order filters are intervals.

Background

The paper points out that Boolean antichains interact with the order filters and order ideals that they generate. Although the paper develops enumeration methods for several families of Boolean antichains, it does not address the subclass for which the associated order filters have the additional property of being intervals.

The question asks for an enumeration of this more restricted class, potentially revealing how the Boolean property constrains the interval structure of generated filters.