Conjecture on simple spectrum of Bethe subalgebras in the specified modules

Let $z_1,\ldots,z_k$ be real parameters satisfying

$$z_1>z_2>\cdots>z_k.$$

Let $L(z_1,\ldots,z_k)$ and $L'(z_1,\ldots,z_k,\cdot)$ denote the modules used in the paper's restriction and multiplicity-space construction. Simple-spectrum conjecture. Under these assumptions, the algebra $\mathcal{B}^-$ acts on $L(z_1,\ldots,z_k)$ with simple spectrum, while $\mathcal{B}^+$ acts with simple spectrum on

$$L'(z_1,\ldots,z_k,1)\oplus L'(z_1,\ldots,z_k,0)$$

or on

$$L'(z_1,\ldots,z_k,\tfrac{1}{2})\oplus L'(z_1-1,\ldots,z_k,\tfrac{1}{2}).$$

The conjecture would make the eigenbasis indexing independent of the path within the corresponding real parameter sector.