Reduced conjecture on conformal radii for non-Bruno rotation numbers

Fix $d\geq 2$. Let $\theta$ be a non-Bruno irrational number with continued-fraction expansion

$$\theta=[u_0;u_1,u_2,\ldots].$$

For each $n$, define

$$\theta_n=[u_0;u_1,\ldots,u_n,2,2,2,\ldots],$$

and let $I_{d-1}[\theta_n]$ be the corresponding family of normalized degree-$d$ polynomials whose relevant critical points lie on the Siegel-disk boundary. Set $M_n$ to be the maximum of $\operatorname{rad}(P)$ over $P\in I_{d-1}[\theta_n]$.

Reduced conjecture.

$$M_n\underset{n\to\infty}{\longrightarrow}0.$$

This is presented as a weaker conjecture motivated by the proof of the paper's main theorem. It is intended as a route toward the optimality of the Bruno condition, and the source gives no resolution status.