Spectrum regularity conjecture for rainbow cycle lengths

Let $S$ be a spectrum of rainbow cycle lengths, namely the set of cycle lengths occurring as rainbow cycles under a fixed edge-coloring. Spectrum regularity conjecture. The asymptotic behavior of $S$ should be classified into exactly one of the following three categories: (a) $S$ contains all sufficiently large numbers; (b) $S$ contains all sufficiently large even numbers; or (c) $S$ contains all sufficiently large numbers congruent to $2\bmod 4$. Equivalently, in terms of monoids, the spectrum should become regular modulo one, two, or four.