Maximum-center conjecture for caterpillars

Prove that if T is a caterpillar, r≥1, and z is a maximum r-center of T, then the neighbor of z has maximum degree.

Background

A maximum r-center is a vertex contained in a maximum intersecting family of independent r-sets through its star. The conjecture seeks a structural description of maximum centers in caterpillars by relating each such center to a neighboring vertex of maximum degree.

References

Zhang, et al. finish by offering the following conjecture.

— A Survey of the Holroyd-Talbot Conjecture  (2501.16144 - Hurlbert, 27 Jan 2025) in Conjecture 5, Section 4