  We consider quantum integrable models solvable by the nested algebraic Bethe
ansatz and possessing $\mathfrak{gl}_3$-invariant $R$-matrix. We study a new
recently proposed approach to construct on-shell Bethe vectors of these models.
We prove that the vectors constructed by this method are semi-on-shell Bethe
vectors for arbitrary values of Bethe parameters. They thus do become on-shell
vectors provided the system of Bethe equations is fulfilled.
