Existence of the asymptotic rainbow saturation density for cycles

Prove that for every integer \(r\ge 5\), the limit \(\lim_{n\to\infty}\operatorname{rsat}(n,C_r)/n\) exists.

Background

The paper proves linear upper and lower bounds for the rainbow saturation number of cycles but does not establish convergence of the normalized quantity. Conjecture 8.1 proposes that, for each fixed cycle length at least five, the edge density of minimally rainbow-saturated graphs has a well-defined asymptotic limit as the number of vertices tends to infinity.