Ferromagnetic Ordering of Energy Levels conjecture

Let $H_V$ be a ferromagnetic Heisenberg Hamiltonian on a finite graph or finite volume, and let $\mathcal{H}^{(S)}$ be its invariant subspace of total spin $S$. Define

$$E(H_V,S):=\min\operatorname{spec}\left.H_V\right|_{\mathcal{H}^{(S)}}.$$

Ferromagnetic Ordering of Energy Levels conjecture. All ferromagnetic Heisenberg models have the FOEL property: if $S'<S$, then

$$E(H_V,S)<E(H_V,S').$$

FOEL orders the lowest energies strictly by decreasing total spin and implies that maximal total spin gives the ground-state energy. The supplied text describes partial results and numerical support, leaving the general assertion open.