Functionally countable Ψ-space with crowded Čech–Stone remainder

Determine whether there exists an Isbell–Mrówka Ψ-space whose space is functionally countable and whose Čech–Stone remainder is crowded (i.e., has no isolated points).

Background

The authors note that if an Isbell–Mrówka Ψ-space has a scattered Čech–Stone remainder, then it yields a functionally countable separable uncountable example. They ask whether such functionally countable Ψ-spaces can have a crowded Čech–Stone remainder, probing the limits of known constructions.

This question aims to clarify whether the scatteredness of the remainder is necessary for functional countability in this class.