Existence in ZFC of a hereditarily separable, exponentially separable uncountable space

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

Background

The space constructed in Theorem 22 is separable and exponentially separable but not hereditarily separable. Under additional set-theoretic assumptions (Ostaszewski space), one can obtain a hereditarily separable exponentially separable uncountable space.

The question seeks a ZFC resolution for the existence of such spaces without additional axioms.

References

  1. Question Is there in ZFC an example of a hereditarily separable, exponentially separable uncountable space?
— Comparing functional countability and exponential separability (2403.15552 - Hernández-Gutiérrez et al., 22 Mar 2024) in Question 25, Section 4