The characterization of Fréchet–Urysohn free topological groups of length four

Let $X$ be a metrizable space, and let $F_4(X)$ denote the corresponding free topological group of length at most four. Suppose that the set of all non-isolated points of $X$ is compact. The characterization conjecture. Then $F_4(X)$ is Fréchet–Urysohn. This fills the missing case between the known characterizations for $F_3(X)$ and $F_n(X)$ for $n\geq 5$; the source presents it as a conjecture posed in earlier work, and no resolution is given here.