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

Ascertain 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 a cosmic space (i.e., has a countable network) under the assumption that C_p(X) is hereditarily sequentially separable.

Background

Velichko proved that every cosmic space is hereditarily sequentially separable, and the paper establishes several equivalences relating hereditary F-separability, σ-separability, and Fréchet–Urysohn properties in various contexts.

This question asks for the converse direction in the field of C_p(X): whether hereditary sequential separability of C_p(X) entails the cosmic property for C_p(X).