Smooth mixed projective curves and a conjecture

Let $f(\mathbf{z},\bar{\mathbf{z}})$ be a strongly mixed homogeneous polynomial of 3 variables $\mathbf{z}=({\mathbf{z}}_{\mathbf{1}},{\mathbf{z}}_{\mathbf{2}},{\mathbf{z}}_{\mathbf{3}})$ of polar degree $q$ with an isolated singularity at the origin. It defines a smooth Riemann surface $C$ in the complex projective space ${\mathbb{P}}^2$. The fundamental group of the complement ${\mathbb{P}}^2\setminus C$ is cyclic group of order $q$ if $f$ is homogeneous polynomial without $\bar{\mathbf{z}}$. We propose a conjecture that this may be even true for mixed homogeneous polynomials by giving several supporting examples.