Completeness of the Bethe Ansatz for the periodic XXZ spin chain

Let $L$) be the length of a one-dimensional periodic Heisenberg–Ising XXZ spin-$\tfrac{1}{2}$ chain, let $N$ be the number of up-spins, and let $\Delta\in\mathbb{R}$ be its anisotropy parameter. For some $\varepsilon(N,L)=\varepsilon>0$, consider anisotropies satisfying $|\Delta|<\varepsilon$. The Bethe Ansatz produces Bethe eigenvectors for this chain.

Completeness conjecture. The Bethe Ansatz completely determines the spectrum for all but finitely many values of $\Delta$ with $|\Delta|<\varepsilon$.

Completeness would establish that, apart from finitely many exceptional anisotropy values in a neighborhood of the isotropic-free point, the Bethe eigenvectors account for the entire spectrum. The source reports extensive numerical evidence but does not provide a proof.