Characterization of weakly aleph-spaces for spaces of continuous functions

Let $X$ be a Tychonoff space. The space $C_c(X)$ denotes $C(X)$ endowed with the topology of uniform convergence on compact subsets of $X$.

Weak-space characterization conjecture. $C_c(X)$ is a weakly $\aleph$-space if and only if $C_c(X)$ is a weakly $\aleph_0$-space if and only if $X$ is a countable $\aleph_0$-space.

The preceding result establishes that $C_c(X)$ is a weakly $\aleph_0$-space for every metrizable and countable space $X$. The conjecture asks whether the stated equivalences hold for all Tychonoff spaces.