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on Haldane gap for the spin-1 Heisenberg chain

The statement

For the spin-1 antiferromagnetic Heisenberg chain $H_L=\sum_{j=1}^L\vec S_j\cdot\vec S_{j+1}$ with periodic boundary conditions and coupling one, prove $\liminf_{L\to\infty}(E_1(L)-E_0(L))>0$ , where $E_1$ is the first energy strictly above the ground space.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.