Non-triviality of Bethe vectors in the Gaudin model

Let $L$ be a tensor product of irreducible finite-dimensional representations of $sl_{N+1}(\mathbb{C})$, marked by complex parameters $z=(z_1,\ldots,z_n)$. A Bethe vector is a vector produced by the Bethe Ansatz in the corresponding Gaudin model. Non-triviality conjecture. In the $sl_{N+1}(\mathbb{C})$ Gaudin model, every Bethe vector is non-zero, for any $z$. Non-vanishing of Bethe vectors is important for identifying Bethe solutions with representation-theoretic and Schubert-calculus data; the source provides no resolution of this assertion.