The polar-set characterization for strongly incomplete polynomial approximation

Let $K\subset {\mathbb{C}}\setminus \{0\}$ be a compact set. A polynomial of the indicated form is $\sum_{k=n/\tau_n}^{n}a_kz^k$, where $\tau_n\to 1$. Polar-set characterization conjecture. Any function continuous on $K$ (or equivalently, any polynomial) can be uniformly approximated on $K$ by polynomials of the form

$$\sum_{k=n/\tau_n}^{n}a_kz^k$$

with $\tau_n\to 1$, if and only if $K$ is polar. This conjecture seeks to characterize exactly the compact sets supporting approximation by strongly incomplete polynomials; the supplied context gives no resolution.