Centrality conjecture for shifted determinants

Let $\lambda$ be a dominant weight, let $V_\lambda$ be the corresponding representation space, and let $\Omega_\lambda(u)=\Omega_\lambda+u\otimes \operatorname{id}$. Using a basis of $V_\lambda$, form the shifted determinant $D_\lambda(u)$ as the column-determinant of the matrix $\Omega_\lambda(u)-L$, where $L=\operatorname{diag}(m,m-1,\ldots,0)$. Shifted-determinant centrality conjecture. For any dominant weight $\lambda$, there exists a basis of $V_\lambda$ such that the coefficients of $D_\lambda(u)$ belong to the center $Z(\mathfrak{gl}_n(\mathbb{C}))$. The paper introduces shifted determinants as a second family of central-polynomial candidates; no resolution of this assertion is supplied in the given text.