Thermodynamic Bethe-root contour conjecture for the homogeneous six-vertex model

Fix the ratio $n/N$, where $n$ is the number of Bethe roots and $N$ is the system size, and let $C$ denote the contour described in the paper. Thermodynamic Bethe-root contour conjecture. As $N\rightarrow\infty$, the roots of the Bethe equations corresponding to the quantum numbers $I_j$ from the Bethe quantum-number conjecture become distributed along the contour $C$.

This conjecture describes the thermodynamic behavior of the Bethe roots associated with the maximal transfer-matrix eigenvalue in the homogeneous case at fixed filling ratio. The contour is subsequently specified in the paper, while a general proof of this limiting root distribution is not supplied here.