Characterize PA for unbounded boundedly compact ultrametric spaces

Prove or disprove that a function f: [0, ∞) → [0, ∞) belongs to P_A if and only if f ∈ P_CU and lim_{t→∞} f(t) = +∞, where A denotes the class of all unbounded boundedly compact ultrametric spaces, P_A := P_{A,A} is the set of A–A preserving functions, and P_CU := P_{CU,CU} is the set of functions that preserve compact ultrametric spaces.

Background

The paper studies functions f: [0, ∞) → [0, ∞) that preserve various classes of ultrametric spaces under composition with a metric d, i.e., (X, d) ↦ (X, f∘d). For a class A of spaces, P_A denotes the set of functions that map every space in A back into A. Earlier results (Theorem 29) identify P_CU with the strongly ultrametric preserving functions (those that preserve topology) and show several equalities among preservers for compact and totally bounded ultrametric spaces.

Here, A is defined as the class of all unbounded boundedly compact (proper) ultrametric spaces. The conjecture proposes a complete characterization of P_A in terms of two conditions: belonging to P_CU and divergence at infinity lim_{t→∞} f(t) = +∞, which is intended to ensure preservation of unboundedness alongside properness.