Rationality conjecture for crossing limits of periodic tiles

Let $T$ be a tile, and let $c_n(T)=\operatorname{cr}(\bigcirc(T^n))$. The limit

$$c(T)=\lim_{n\to\infty}\frac{c_n(T)}{n}$$

exists. Rationality conjecture for crossing limits. There exists a computable function $f$ that assigns to every tile $T$ a positive integer $f(T)$ such that $c(T)$ is rational with denominator at most $f(T)$. The existence of such a bound would establish a strong form of eventual periodicity for optimal drawings and make the asymptotic crossing rate effectively rational; this remains open.