Intersection alternatives for six-vertex Bethe contours

Let $C$ be the smooth Bethe-root curve and define

$$C_1=\left\{z=\frac{1}{2\Delta-w}:w\in C\right\},\qquad C_2=\left\{z=-\frac{1}{w}+2\Delta:w\in C\right\}.$$

Assume $\Delta\neq0$, and let $\alpha=n/N$ be fixed. Bethe-contour intersection conjecture. Exactly one of the following occurs: $C$, $C_1$, and $C_2$ do not intersect; or all three intersect at the two points $w_\pm=e^{\pm i\eta}$, which solve $w^2-2\Delta w+1=0$ and are conjugate, with the points possibly coinciding in the degenerate case. The second alternative is possible only when $\alpha=1$. The source does not give a resolution for general $\Delta\neq0$.