Bethe quantum-number conjecture for the maximal six-vertex-model eigenvalue

Let $n$ be the fixed number of Bethe roots, let $H$ be the parameter in the transfer matrix, and let $I_j$ denote the corresponding Bethe quantum numbers. Bethe quantum-number conjecture. When the inhomogeneities and $H$ are sufficiently small, or when the system is homogeneous, the maximal eigenvalue corresponds to solutions of the Bethe equations with

$$I_j=\frac{n+1-2j}{2},\qquad j=1,\ldots,n.$$

This conjecture identifies the Bethe state describing the largest eigenvalue in the fixed-$n$ subspace. It has a long history in the Bethe-ansatz analysis of the Heisenberg XXZ chain and the six-vertex model, and is used to characterize the ground state and compute the thermodynamic free energy; the stated finite-$N$ claim remains presented as a conjecture here.