Ultrametric approximation conjecture for compact semimetric spaces

Let $X$ be a compact nonempty semimetric space and let $\varepsilon>0$. A compact semimetric approximation conjecture asserts that there exist continuous mappings

$$F\colon Y \to X,\qquad \Phi\colon W \to Y$$

such that $Y$ is compact ultrametric, $W\in\mathfrak U_C$, $\Phi$ is an $\varepsilon$-isometry, and $F$ is ball-preserving. Moreover,

$$\mathbf{B}_Y\ni B\mapsto F(B)\in\mathbf{B}_X$$

is an arc-surjective homomorphism from $(\mathbf{B}_Y,A_Y)$ to $(\mathbf{B}_X,A_X)$. Here $\mathfrak U_C$ denotes the class of compact ultrametric spaces all of whose finite subspaces belong to $\mathfrak U$, the class of finite ultrametric spaces extremal for the Gomory–Hu inequality. This conjecture proposes that every compact nonempty semimetric space admits arbitrarily accurate structural approximation by a compact ultrametric space; the supplied text gives no resolution, so its status remains open.