I have a martingale difference series $(Z_t)$ with respect to a given filtration $(\mathcal{F}_t)$, i.e. $E[Z_t \mid \mathcal{F}_{t-1}] = 0$. Furthermore, $E[Z_t^2 \mid \mathcal{F}_{t-1}] = 1$. I also have a process $$\sigma_t^2 = \alpha + \phi\sigma_{t-1}^2 + \theta X_{t-1}^2$$ and $$X_t = \sigma_t Z_t$$ We have $\sigma_t$ is measurable with respect to $\mathcal{F}_{t-1}$. Now suppose $(X_t)$ is a stationary process. Then, $$E[X_t^2\mid\mathcal{F}_{t-1}] = E[\sigma_t^2Z_t^2\mid\mathcal{F}_{t-1}]=\sigma_t^2E[Z_t^2\mid\mathcal{F}_{t-1}] = \sigma_t^2$$ Hence, $E[X_t^2] = E[\sigma_t^2]$. Since $X_t$ is stationary and has zero mean, $E[X_t^2]$ and therefore, $E[\sigma_t^2]$ is independent of $t$.
Is it then also true that $E[\sigma_t^4]$ is independent of $t$?