Bethe-root condensation on a symmetric curve

Let $N\to\infty$ and $n\to\infty$ with $\alpha=n/N$ fixed. Consider the Bethe equations for the six-vertex transfer matrix and the maximal eigenvalue in the subspace $(\mathbb{C}^{\otimes N})_n$. Bethe-root condensation conjecture. The corresponding roots concentrate on a smooth curve $C$ in the complex $z$-plane, and $C$ is symmetric under complex conjugation $z\mapsto\bar z$. These conjectures are stated as supported by theoretical and numerical results and are provable in the free-fermion case $\Delta=0$; the general case remains open.