Lack of Gevrey solvability for a model operator

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