The ground-state energy ordering conjecture for Heisenberg XXZ and XY models

Let $L$ be the sector label and let $\varepsilon_m$ denote the quantity introduced in the paper for sector $m$. Ground-state energy ordering conjecture.

$$\varepsilon_m\leq\varepsilon_L$$

for all $m$ such that $-L\leq m<L$. This inequality is the remaining unproved condition needed to verify the stated assumption; the paper provides numerical and heuristic justification but no analytic proof.