A Kohn-nirenberg example using lower degree

We will construct polynomials of degree 6 in z and $\bar{z}$ on $C^2$ which gives, via its coefficient $\beta$ as a parameter, a family of pseudoconvex domains $\Omega_\beta$ in $C^2$ with the origin being a boundary point, and show that the domains $\Omega_\beta$ has no peak functions of class $c^1$ at the origin and has no holomorphic support functions for $1 \leq \beta < \frac{9}{5}$.