Does hereditarily separable imply hereditarily 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 hereditarily sequentially separable under the assumption that C_p(X) is hereditarily separable.

Background

The paper studies relationships between sequential separability, σ-separability, F-separability, and their hereditary variants in function spaces C_p(X). Several equivalences are established under additional structural conditions on X (e.g., γ-space) and on powers of X, but the general implication from hereditary separability to hereditary sequential separability is not resolved.

Hereditarily separable means every subspace of C_p(X) is separable; hereditarily sequentially separable requires every subspace to be sequentially separable. The question asks whether the stronger sequential property follows from hereditary separability in the C_p setting.