Huang's conjecture on balanced cyclic $(132,213)$-avoiding permutations

Let $\mathcal C_n(132,213)$ be the set of cyclic permutations of size $n$ avoiding the patterns $132$ and $213$, and let $\mathcal C_n^B(132,213)$ be the subset of balanced such permutations. Huang's conjecture. For even $n$,

$$\frac{|\mathcal C_n^B(132,213)|}{|\mathcal C_n(132,213)|}=\Omega(1).$$

The conjecture predicts that balanced cyclic permutations form a positive proportion, up to a constant lower bound, of all cyclic $(132,213)$-avoiding permutations along the even sizes. The paper presents it as a possible direction for obtaining a general lower bound; its resolution is not given here.