The characterization conjecture for infinite trees with finite locating chromatic number

Let $T$ be an infinite tree with bounded degree. Define $f(T)$ by removing all end-paths of $T$ except the end-branches, and for $n\geq 2$ set $f^n=f^{n-1}\circ f$. A path may be infinite, finite, or equal to $K_1$. Characterization conjecture. The locating chromatic number of $T$ is finite if and only if there is an integer $n$ such that

$$f^n(T)=P,$$

where $P$ is a path. The conjecture's reverse implication is proved in the source; the forward implication is supported by examples but remains open.