Infinite family of minimal two-ended locally finite trees with localization number 2

Determine whether there exists an infinite family of minimal locally finite trees with exactly two ends and localization number 2, where minimal means that deleting any single edge reduces the localization number to 1.

Background

The paper studies the localization number ζ(G) on locally finite graphs and, in particular, on trees. For finite trees, ζ(T) ≤ 2, but the authors show that locally finite trees can have a wide range of localization numbers. They prove that trees with countably many ends have ζ(T) ≤ 2, yet give examples showing that the obstruction to ζ(T) = 1 is more complex than in the finite case.

They present the doubly infinite comb tree as a T-free locally finite example with ζ = 2 and show it is minimal in the sense that removing any edge drops the localization number to 1. Motivated by this, they pose the problem of determining minimal trees with ζ = 2 and, specifically, whether there is an infinite family of minimal two-ended trees with ζ = 2.