Does F-separable imply Fréchet–Urysohn 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, has the Fréchet–Urysohn property under the assumption that C_p(X) is F-separable.

Background

The Fréchet–Urysohn property for C_p(X) is deeply connected to the γ-property of X (Gerlits–Nagy). The paper establishes equivalences between hereditary F-separability and Fréchet–Urysohn plus hereditary separability under certain conditions, but the general implication from F-separability to Fréchet–Urysohn remains open.

This question seeks to clarify whether F-separability alone guarantees the sequence-based convergence characterization in C_p(X).