Graphs that are not complete pluripolar

Let $D_1\subset D_2$ be domains in $\mathbb{C}$. Under very mild conditions on $D_2$ we show that there exist holomorphic functions $f$, defined on $D_1$with the property that $f$ is nowhere extendible across $\partial D_1$, while the graph of $f$ over $D_1$ is not complete pluripolar in $D_2\times\mathbb{C}$. This refutes a conjecture of Levenberg, Martin and Poletsky (1992).