The Riccati-eigenfunction rotation-number conjecture

Let $l\in\mathbb N$, and let the parameter $\mu$ satisfy one of the equations governing the relevant monodromy cases according as $l$ is odd or even. Consider the corresponding monodromy eigenfunction of the Heun equation with eigenvalue $-1$. Riccati-eigenfunction rotation-number conjecture. The corresponding solution of the Riccati equation gives a periodic solution of the associated Josephson equation on the two-torus whose rotation number lies between $0$ and $l$. This conjecture is proposed as a strategy for locating intersections of phase-lock-area boundaries with specified parameter lines; the source gives no resolution.