The spectral-curve uniqueness conjecture for generalized simple intersections

Fix $\ell\in\mathbb Z_{\neq 0}$ and $\omega>0$. Let $\Gamma_\ell\subset\mathbb R^2_{(\lambda,\mu)}$ be the real spectral curve of the special double confluent Heun equations, and impose

$$\lambda=\frac{1}{4\omega^2}-\mu^2.$$

The corresponding parameters are $(B,A)=(\ell\omega,2\mu\omega)$, with rotation number $\rho(B,A)$. Spectral-curve uniqueness conjecture. For every $\omega>0$, the real spectral curve $\Gamma_\ell$ contains a unique point $(\lambda,\mu)$, up to changing the sign of $\mu$, satisfying the displayed relation and

$$\rho(\ell\omega,2\mu\omega)=\ell.$$

The point corresponds to the higher generalized simple intersection $\mathcal P_\ell$ and this conjecture is stated as an algebro-geometric equivalent of the connectivity conjecture; the source gives no evidence that it has been resolved.