Square of subgaussian variable's parameters

Question

If a zero-mean variable $X$ satisfies $\mathbb{E}[e^{\lambda X} ] \le e^{\lambda^{2} \sigma^{2}/2}$ for all $\lambda \in \mathbb{R}$, (i.e., subgaussian with parameter $\sigma^{2}$), does it follow that $\mathbb{E}[e^{\lambda (X^{2} - \mathbb{E}[X^{2}])}] \le e^{\lambda^{2}\sigma^{4}/2}$?