The finiteness conjecture for large Fatou components of polynomials

Let $f:\mathbb{C}\to\mathbb{C}$ be a polynomial, and let a Fatou component mean a connected component of the Fatou set of $f$. For $\epsilon>0$, measure diameters in the usual Euclidean metric on $\mathbb{C}$. Finiteness conjecture. For every $\epsilon>0$, there are at most finitely many Fatou components of $f$ with diameters greater than $\epsilon$. This conjecture proposes that Fatou components of a polynomial become uniformly small apart from finitely many exceptions at every positive scale; the surrounding discussion highlights difficulties caused by irrationally neutral cycles and Cremer points, and no resolution is supplied here.