The dilation asymptotic conjecture for compact sets

Let $K$ be a non-empty compact subset of the complex plane, and let $\delta_n$ and $C$ denote the quantities defined in the preceding results. Dilation asymptotic conjecture.

$$\lim_{n\to\infty}\frac{\delta_n\left(K^{\frac{1}{n}}\right)}{C\left(K^{\frac{1}{n}}\right)}=1.$$

This would improve the preceding result on dilation; by the Fekete–Szegő theorem, the assertion already holds when $K$ is non-polar.