Cyclic-vector conjecture for Bethe subalgebras in tame Yangian representations

Let $X\in\overline{M_{0,n+2}}$ and let $B(X)$ be the corresponding Bethe subalgebra of $Y(\mathfrak{gl}_n)$. A representation of $Y(\mathfrak{gl}_n)$ is tame if it has the form $\bigotimes_{i=1}^k V_{\lambda_i\setminus\mu_i}(z_i)$ with $z_i-z_j\notin\mathbb{Z}$ for $i\ne j$. A vector $v$ is cyclic for $B(X)$ if $B(X)\cdot v=V$. Cyclic-vector conjecture. $B(X)$ has a cyclic vector in every tame representation of $Y(\mathfrak{gl}_n)$ for every $X\in\overline{M_{0,n+2}}$. The conjecture concerns the first step toward proving that Bethe-ansatz eigenvectors form a basis: cyclicity prevents multiplicities among joint eigenvalues, while the assertion is not established in the stated generality.