Coordinate-basis expansion and linear independence of Bethe vectors

Let $\mathbb{V}$ be the coordinate-basis state space, let $\mathcal{X}$ be its set of coordinate configurations, and let $\Xi$ be the set of Bethe-root solutions modulo the relevant permutation equivalence. For $\mathbf{x}\in\mathcal{X}$, write $|\mathbf{x}\rangle\in\mathbb{V}$ for the coordinate basis element. For $\boldsymbol{\xi}\in\Xi$, let $|\boldsymbol{\xi}\rangle\in\mathbb{W}$ be the corresponding Bethe vector, and let $\ell(\mathbf{x},\boldsymbol{\xi})$ be the coefficient map defined in the source.

Bethe-vector basis conjecture. For every $|\mathbf{x}\rangle\in\mathbb{V}$,

$$|\mathbf{x}\rangle=\sum_{[\boldsymbol{\xi}]\in\Xi}\ell(\mathbf{x},\boldsymbol{\xi})|\boldsymbol{\xi}\rangle.$$

Moreover, the set $\{|\boldsymbol{\xi}\rangle\mid[\boldsymbol{\xi}]\in\Xi\}$ is linearly independent.

This claim would give an explicit inverse transformation from the Bethe-vector, or energy, basis to the coordinate basis and would prove completeness constructively. The source proves the claim at $\Delta=0$ and reports numerical verification for other values of $\Delta$, but does not establish it in general.