Caterpillar maximum-center conjecture

Prove that, for every caterpillar, every r ≥ 1, and every maximum r-center z, the neighbor of z has maximum degree.

Background

The paper surveys results showing that certain tree families have maximum centers at leaves or pendant vertices, while other trees can have maximum centers far from leaves. For caterpillars, Zhang, Ran, and Huang formulate a structural conjecture relating any maximum center to the degree of its neighbor.