The quadratic-differential characterization of the support

For a parameter $b$, let $\Theta$ be the variable in the quadratic differential

$$\Psi_{b,\Lambda}=-\frac{\Theta(\Theta+b)^2-4\Theta-\Lambda}{\Theta}\,d\Theta^2,$$

where $\Lambda$ is a complex parameter. Quadratic-differential support conjecture. The support of $\mu_b$ coincides with the set of values of $\Lambda$ such that the differential $\Psi_{b,\Lambda}$ has two critical horizontal trajectories. The preceding analysis identifies the relevant differential and motivates this characterization through the geometry of its zeros and pole, but the source supplies no proof or resolution of the claim.