Does hereditarily separable imply sequentially separable for C_p(X)?

Determine whether the function space C_p(X), consisting of all real-valued continuous functions on a Tychonoff space X endowed with the topology of pointwise convergence, is sequentially separable under the assumption that C_p(X) is hereditarily separable.

Background

This question is identified by the authors as a weakening of their Question 4.1. While the paper establishes several equivalences among hereditary properties under additional assumptions on X, the general implication from hereditary separability to sequential separability is left unresolved.

Sequential separability requires the existence of a countable set whose sequential closure is the entire space, a stronger requirement than mere separability.