Gaplessness of half-integer spin Heisenberg chains with $S \ge 3/2$

For $H = \sum_i S_i . S_{i+1}$ with $S = 3/2$ (or any half-integer $S \ge 3/2$), prove that $E_1(L) - E_0(L) \to 0$ as $L \to \infty$ with a unique ground state, or equivalently exclude dimerization. The Lieb-Schultz-Mattis theorem (and Affleck-Lieb extension) gives gapless or degenerate. Answer: a proof.

A proof that the $S = 3/2$ chain is gapless with a unique ground state.