Existence in ZFC of a perfectly normal, separable, exponentially separable uncountable space

Determine whether, within ZFC, there exists an uncountable topological space that is perfectly normal, separable, and exponentially separable.

Background

The authors construct an uncountable separable exponentially separable space but note it is not perfectly normal. Under additional set-theoretic assumptions (Ostaszewski space), they obtain an example that is perfectly normal, hereditarily separable, and exponentially separable.

They ask whether such an example can be obtained in ZFC, highlighting a gap between consistency results and ZFC constructions.