The real spectral-curve component conjecture

For each $\ell\in\mathbb N$, let $\Gamma_\ell$ be the real spectral curve defined by the polynomial $Q_\ell(\lambda,\mu^2)$, and consider its intersection with the domain $\{\lambda+\mu^2>0\}$. Let $\Gamma_\ell^*$ denote the connected component whose points correspond to the differential equations in question on $\mathbb T^2$ with rotation number $\ell$.

Real spectral-curve component conjecture. The following assertions hold:

1. The intersection contains a unique such connected component $\Gamma_\ell^*$. Every other component corresponds to a positive rotation number less than $\ell$.
2. The projection

$$(\lambda,\mu)\longmapsto\lambda+\mu^2$$

maps $\Gamma_\ell^*$ diffeomorphically onto $\mathbb R_+$.

This is presented as a slightly stronger reformulation of a pure real-algebro-geometric conjecture and as a reduction of the connectivity conjecture. The supplied text does not state that it has been resolved.