Weakly First Countable Compact Hausdorff Spaces That Are Not First Countable

A topological space $X$ is called weakly first countable if there exists a function $V : X \to \mathbb{N} \to \mathcal{P}(X)$ such that: (i) for every $x \in X$ and $n \in \mathbb{N}$, we have $x \in V(x,n)$; (ii) for every $x \in X$ and $n \in \mathbb{N}$, $V(x,n+1) \subseteq V(x,n)$; and (iii) a subset $O \subseteq X$ is open if and only if for every $x \in O$ there exists $k \in \mathbb{N}$ such that $V(x,k) \subseteq O$. A topological space is first countable if every point has a countable neighborhood basis. Does there exist in ZFC a weakly first countable compact Hausdorff space that is not first countable?