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

Background

The paper shows that σ-separability coincides with sequential separability for C_p(X), and studies F-separability and its hereditary variants. While σ-separability and sequential separability coincide, it remains unresolved whether sequential separability implies F-separability in C_p(X).

F-separability requires the existence of a countable set whose F-closure is the entire space, a stronger structural condition than σ-separability.