Random Runge approximation conjecture for holomorphic functions on compact sets

Let $K\subset\mathbb C$ be compact, and let $f(\omega,z)$ be a random holomorphic function on $K$. A sequence $R_k(\omega,z)$ of random rational functions should exist such that, for each $\omega$,

$$R_k(\omega,\cdot)\rightarrow f(\omega,\cdot)$$

uniformly on $K$.

Random Runge approximation conjecture. There is a sequence $R_k(\omega,z)$ of random rational functions such that, for each $\omega$, $R_k(\omega,\cdot)\rightarrow f(\omega,\cdot)$ uniformly on $K$.

This is proposed as a stronger version of the preceding random Runge theorem, which assumes that $f(\omega,\cdot)$ is holomorphic on an open neighborhood $U$ of $K$ for every $\omega$. The conjecture seeks approximation under the weaker assumption of holomorphy on the compact set itself; its resolution is not given here.