Douady's optimality conjecture for the Bruno condition

Let $\theta\in\mathbb{R}$ be a rotation number, and call it a Bruno number when its Bruno sum $B(\theta)$ is finite. A polynomial has a Siegel disk of rotation number $\theta$ if it has an invariant disk on which its dynamics is analytically conjugate to the rotation by $\theta$.

Douady's optimality conjecture. If $\theta$ is not a Bruno number, then no polynomial of degree at least $2$ can have a Siegel disk of rotation number $\theta$.

The conjecture identifies the Bruno condition as necessary for the existence of Siegel disks. Its special case for degree $2$ was proved by Yoccoz, while the general statement is recalled here as open.