Eventual odd circular mixing conjecture for bipartite graphs

Let $G$ be a bipartite graph. A graph is $(p,q)$-mixing when it has the circular mixing property with parameters $p$ and $q$.

Eventual odd circular mixing conjecture. There exists an integer $q_0$ such that, for every integer $q\geq q_0$, the graph $G$ is $(2q+1,q)$-mixing.

If true, this would show that every bipartite graph has circular mixing threshold at most $2$, addressing the paper's observation that no bipartite graph is currently known with circular mixing threshold greater than $2$.