A question on the Cauchy problem in the Gevrey classes for weakly hyperbolic equations

For a homogeneous polynomial $p$ in $\xi \in {\mathbf{R}}^n$ with Gevrey coefficients, it is known that the Cauchy problem for any realization of $p$ is well-posed in the Gevrey class of order $s<2$ if the characteristic roots are real. In this note, we give examples showing the situation of the converse direction, in particular the optimality of the Gevrey order $s=2$.