The balanced cyclic composition proportion conjecture

Let $C_n(132,213)$ denote the number of cyclic compositions of $n$ avoiding the patterns $132$ and $213$, and let $C^B_n(132,213)$ denote the number of balanced cyclic compositions of $n$ (equivalently, balanced reverse layered permutations of length $n$) avoiding these patterns. Here $\frac{C^B_n(132,213)}{C_n(132,213)}$ is considered along the even positive integers $n$.

Balanced cyclic composition proportion conjecture. For even $n$,

$$\frac{C^B_n(132,213)}{C_n(132,213)} = \Omega(1).$$

The conjecture asserts that balanced cyclic compositions comprise a nonvanishing proportion of all $(132,213)$-avoiding cyclic compositions as the even size grows. The source presents this as a conjecture suggested by computer data, and no resolution is supplied.