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 $p\in \widehat{K}\setminus K$ such that there does not exist a sequence $\{A_{\nu }\}$ of analytic sets $A_{\nu }\subset \subset {\mathbb{C}}^2$ with boundary satisfying $p\in A_{\nu }$ for every $\nu \in \mathbb{N}$ and ${\lim }_{\nu \to \mathrm{\infty }}b{A}_{\nu }\subset K$. For every point in $\widehat{K}\setminus K$, we explicitly construct a sequence of Poletsky discs, and we compute the weak limit of the pushforwards of the Green current under these discs.