The unique-arc characterization of lion-winning Euclidean metric spaces

Let $X$ be a subset of a Euclidean space equipped with an associated path length metric, meaning that any two points are joined by a path of finite length, and suppose that $X$ is compact in this metric. An arc is an injective path.

Unique-arc conjecture. The lion has a winning strategy in the lion and man game on $X$ from all starting positions if and only if any two points of $X$ are joined by a unique arc.

This is proposed as a stronger statement describing exactly when the lion wins in the bounded-time game on compact, path-metric subsets of Euclidean space. The surrounding discussion presents it as an example of the expected tree-like or dendrite-like characterization; no resolution is supplied here.