Approximation of polynomial hulls by analytic varieties: A counterexample

We construct a connected, compact set $K \subset \mathbb{C}^2$ with the following property: there exist points $p \in \hat{K} \setminus K$ such that there does not exist a sequence $\{A_ν\}$ of analytic sets $A_ν\subset\subset \mathbb{C}^2$ with boundary satisfying $p \in A_ν$ for every $ν\in \mathbb{N}$ and $\lim_{ν\to\infty} bA_ν\subset K$. For every point in $\hat{K} \setminus K$, we explicitly construct a sequence of Poletsky discs, and we compute the weak limit of the pushforwards of the Green current under these discs.