Lavrentiev–Andersson zero-free polynomial approximation conjecture

Let $K$ be a compact set with connected complement, and let $f$ be a continuous function on $K$ that is analytic in the interior $K^\circ$ and zero-free on $K^\circ$. Lavrentiev–Andersson conjecture. For every $\varepsilon>0$, there exists a polynomial $p$ that is zero-free on $K$ and satisfies

$$\max_{z\in K}|f(z)-p(z)|<\varepsilon.$$

This conjecture was motivated by zero-free versions of Lavrentiev's theorem and by the Voronin universality theorem; the source states that it has since been proved in increasing generality.