Ordinary amplitude support can miss a singular-support limit

For $\Phi (s,t,\theta )=s\theta$, take a smooth symbol that turns on increasingly high frequencies in disjoint spatial bumps centered at $t_j\to 0$. Superpolynomially small bump coefficients keep all symbol class estimates valid. The amplitude vanishes near every finite-frequency point over $t=0$, yet its oscillatory integral has delta singularities at $(0,t_j)$. Closedness of singular support forces a singularity at $(0,0)$ as well. A bound using only ordinary, rather than closed conic, amplitude support can consequently fail.