On Newton Diagrams of Plurisubharmonic Polynomials

Each extreme edge of the Newton diagram of a plurisubharmonic polynomial on ${\mathbb{C}}^2$ gives rise to a plurisubharmonic polynomial. It is tempting to believe that the union of the extreme edges or the convex hull of said union will do the same. We construct a plurisubharmonic polynomial $P$ on ${\mathbb{C}}^2$ with precisely two extreme edges $E_1$ and $E_2$, such that neither $E_1\cup E_2$ nor $\text{Conv}(E_1\cup E_2)$ yields a plurisubharmonic polynomial.