On the conditions of proportional local assignability of the Lyapunov spectrum of a linear discrete-time system

We consider a problem of assigning the Lyapunov spectrum for a linear control discrete-time system
\begin{equation} x(m+1)=A(m)x(m)+B(m)u(m),\quad m\in\mathbb N,\ x\in\mathbb R^{n},\ u\in\mathbb R^{k}, \tag{1} \end{equation}
in a small neighborhood of the Lyapunov spectrum of the free system
\begin{equation} x(m+1)=A(m)x(m),\quad m\in\mathbb N,\ x\in\mathbb R^{n}, \tag{2} \end{equation}
by means of linear feedback $u(m)=U(m)x(m)$. We assume that the norm of the feedback matrix $U(\cdot)$ satisfies the Lipschitz estimate with respect to the required shift of the Lyapunov spectrum. This property is called proportional local assignability of the Lyapunov spectrum of the closed-loop system
\begin{equation} x(m+1)=\bigl(A(m)+B(m)U(m)\bigr)x(m),\quad m\in\mathbb N,\ x\in\mathbb R^{n}. \tag{3} \end{equation}
We previously proved that uniform complete controllability of system (1) and stability of the Lyapunov spectrum of free system (2) are sufficient conditions for proportional local assignability of the Lyapunov spectrum of closed-loop system (3). In this paper we give an example demonstrating that these conditions are not necessary.