Andersson–Gauthier conjecture on zero-free polynomial approximation

Let $K$ be a compact subset of the complex plane, and write $K^\circ$ for its interior. A continuous function $f:K\to\mathbb C$ is zero-free on $K^\circ$ when $f(z)\neq 0$ for every $z\in K^\circ$. Assume that the complement $\mathbb C\setminus K$ is connected. Andersson–Gauthier conjecture. Every continuous function $f:K\to\mathbb C$ that is holomorphic and zero-free on $K^\circ$ is a uniform limit of polynomials that are zero-free on all of $K$. This is a zero-free refinement of Mergelyan's polynomial approximation theorem; the source presents it as a conjecture of Johan Andersson and Paul Gauthier, and no resolution is supplied here.