Lack of Gevrey solvability for a model operator

We prove that the Cauchy problem for the model hyperbolic operator in $\R^{4}$ $$Q=-D_t^2+2xD_tD_y+D_x^2+x^3D_y^2+D_z^2+z^2D_y^2$$ is not locally solvable at the origin, in the Gevrey $s$ class if $s>6$.