Recall that a topological space is called Frechet-Urysohn if the operations of closure and sequential closure coincide.

Let $X$ be a locally compact Hausdorff space. It is known that $C(X)$ is not Frechet-Urysohn with respect to the compact-open topology. Indeed, it was proven in the paper *Hernández, Mazón - On the sequential spaces of continuous functions*, which I cannot access, that if $X$ is first countable, then $C(X)$ is Frechet-Urysohn iff $X$ is hemi-compact (which means that there is a sequence of compact sets such that every compact is contain in an element of the sequence). My question however is not about the whole $C(X)$, but its subset - $C_{0}(X)$.

Recall that $C_{0}(X)$ is the set of all $f\in C(X)$ that "vanish at infinity", i.e. for every $\varepsilon>0$ there is a compact $K\subset X$ such that $|f(x)|<\varepsilon$ as soon as $x\not\in K$ (equivalently, $f$ is continuously extended to the one-point compactification of $X$ by setting $f(\infty)=0$).

Is $C_{0}(X)$ always Frechet-Urysohn with respect to the compact-open topology (when $X$ is locally compact)?