Characterize PB for uniformly discrete unbounded locally finite ultrametric spaces

Prove or disprove that a function f: [0, ∞) → [0, ∞) belongs to P_B if and only if f ∈ PU and lim_{t→∞} f(t) = +∞, where B denotes the class of all uniformly discrete unbounded locally finite ultrametric spaces, P_B := P_{B,B} is the set of B–B preserving functions, and PU is the set of ultrametric-preserving functions.

Background

The class B consists of ultrametric spaces that are uniformly discrete (there exists ε>0 such that all distinct points are at distance at least ε), unbounded, and locally finite (every bounded subset is finite). The paper characterizes several preserver classes for compact, totally bounded, and non-uniformly discrete ultrametric spaces, but an explicit description of P_B remains unsettled.

This conjecture posits that preserving B is equivalent to being ultrametric-preserving together with the growth condition lim_{t→∞} f(t) = +∞, which is naturally linked to preserving unboundedness in the ultrametric context.