Characterization of ultrametric-preserving functions for unbounded proper ultrametric spaces

Let $f: [0,\infty) \to [0,\infty)$ be a function. Let $\mathbf{A}$ be the class of all unbounded boundedly compact (proper) ultrametric spaces, and let $\mathbf{P}_{\mathbf{A}}$ denote the class of functions preserving the ultrametric property on every space in $\mathbf{A}$. Let $\mathbf{P}_{\mathbf{CU}}$ denote the class of functions preserving the ultrametric property on compact ultrametric spaces. The characterization conjecture. A function $f$ belongs to $\mathbf{P}_{\mathbf{A}}$ if and only if $f \in \mathbf{P}_{\mathbf{CU}}$ and

$$\lim\limits_{t \to +\infty} f(t) = +\infty.$$

The conjecture asks whether preservation on compact ultrametric spaces, together with divergence at infinity, exactly characterizes preservation on all unbounded proper ultrametric spaces. The source presents this as one of two conjectures and gives no resolution.