The focus conjecture for the support of the algebraic spectrum

Let $b=u+iv$ with $v\ne 0$, and let $\Gamma_b$ be the real rational cubic with a node at the origin defined by

$$\left(x-\frac{uy}{v}\right)^2=\frac{(4v-y)y^2}{v^3},$$

where $x=\operatorname{Re}z$ and $y=\operatorname{Im}z$. Its three foci are the origin and the two values

$$f_{1,2}=\frac{2}{27}\left(36b-b^3\pm\sqrt{(12+b^2)^3}\right).$$

Let $\mu_b$ denote the measure whose support is being considered. Focus conjecture. Depending on the value of $b$, the endpoints of the support of $\mu_b$ are either all three foci of $\Gamma_b$ or just two of them, always including the focus at the origin. This predicts that the spectral-support endpoints are selected from the foci of the associated cubic; the source reports numerical support for the claim, but gives no resolution.