Monotonicity of the equal-speed cop number

Prove that for every graph G and all positive integers s<s′, the equal-speed cop number satisfies c_{s,s}(G)≥c_{s′,s′}(G).

Background

The paper proves a factor-monotonicity result: c_{s,s}(G)≥c_{ks,ks}(G) for every positive integer k. It also constructs graphs realizing every nonincreasing positive-integer sequence that eventually reaches 1 as their sequence of equal-speed cop numbers. Despite this evidence, the authors do not prove monotonicity for arbitrary speeds and state it as a conjecture.