Moment generating function of squared norm of multivariate Gaussian

Question

Let $X \in \mathbb{R}^d$ be a zero-mean multivariate Gaussian, with independent components, i.e., $X \sim \mathcal{N}(0,\Sigma)$, with $\Sigma = diag(\sigma_1^2,\ldots,\sigma_d^2)$ a diagonal matrix. Leveraging component independence and (sub-)Gaussianity, it can then be shown that, for any $0 < \lambda \leq \frac{1}{\sigma_{\max}^2}$, we have (up to global constants) $$\mathbb{E}[\exp(\lambda\|X\|^2)] \leq \exp(\lambda tr(\Sigma)),$$ where $\sigma_{\max}^2 = \max_{\{i = 1,\ldots,d\}}\sigma_i^2$ and $tr(\cdot)$ is the trace operator. My question is: can a similar result (e.g., depending on the trace of $\Sigma$) be provided in the case where the components of $X$ are not necessarily independent, i.e., $\Sigma$ is a general positive definite matrix? The classical (sub-)Gaussianity argument will provide a bound that depends only on $\sigma_{\max}^2$, however, I wonder if a tighter dependence is possible?