A two-dimensional counterexample for Kobayashi completeness of balanced domains

Jarnicki and Pflug constructed a bounded balanced domain of holomorphy in ${\mathbb{C}}^n$, $n\ge 3$, with continuous Minkowski function which is not Kobayashi complete, and explicitly asked whether the same phenomenon can occur in dimension two. We give such a domain in ${\mathbb{C}}^2$.