Monotonicity of blue-vertex proportions

Prove or disprove whether the asymptotic proportion of blue vertices in admissibly antichain-colored rooted plane trees with at most k children per vertex is monotone in k, analogously to the proven monotonicity of the red-vertex constants c_k.

Background

For rooted plane trees with maximum outdegree k, the paper proves that the limiting red-vertex proportions c_k increase strictly with k, from 1/3 for k=2 toward 2/5 for unrestricted rooted plane trees. The corresponding blue-vertex proportions are computed for several families, but no monotonicity theorem is established. Numerical evidence suggests that an analogous result may hold, yet the authors state that they were unable to prove it.