Possible numbers of projective zeros of a mixed homogeneous polynomial

Let $q$ and $r$ be nonnegative integers, let $f\in\mathcal M(q+2r,q;2)$, and let

$$V=\{[\mathbf z]\in\mathbb{C}\mathbb{P}^1\mid f(\mathbf z,\overline{\mathbf z})=0\}.$$

Write $\alpha=\sharp V$ for the number of points of $V$. The possible-zero-count claim. The possible values of $\alpha$ are exactly

$$\{q,q+2,\dots,q+2r\}.$$

The preceding proposition constructs examples attaining every listed value, while the exactness of the list is asserted here; the supplied text does not establish whether this claim has been resolved.