Completeness and faithfulness conjecture for Wronskian Bethe equations

Let $\mathcal{H}$ be a cyclic representation of the Yangian $\mathcal{Y}(\mathfrak{so}_{2r})$, with highest-weight vector a cyclic vector. Let $d_{\lambda}\nu_{\lambda}\subset\mathcal{H}$ be the highest-weight subspace with respect to the $\mathfrak{so}_{2r}$ action. Define the Bethe algebra as the commutative subalgebra of $\operatorname{End}(\mathcal{H})$ generated by row-to-row transfer matrices in all finite-dimensional auxiliary representations with periodic boundary conditions, and let $\mathfrak{B}$ be its restriction to this highest-weight subspace. Let $c_{a,l}$, for $a=0,\ldots,r$, satisfy the finite-difference Wronskian, Wronskian, and spinor-from-vector conditions, together with the prescribed gauge-fixing conditions for $\psi_a$ and $\psi_{ab}$; these variables form the Wronskian Bethe algebra $\mathcal{W}$. Completeness and faithfulness conjecture. The algebra $\mathfrak{B}$ is a maximal commutative subalgebra of the endomorphisms of the highest-weight subspace, and $\mathcal{W}$ is isomorphic to $\mathfrak{B}$. In particular, the algebraic number of solutions of the stated equations with the prescribed gauge fixing is $d_{\lambda}$. Although the formalism was tested successfully in numerous examples, including nongeneric inhomogeneities and reducible representations, the claim is presented as an experimental conjecture; a general proof of completeness and faithfulness remains open.