Fix a non-negative integer $k$ and let $M^1:\mathbb{R}^n\rightarrow \mathbb{R}^n$ and $M^2,\Sigma:\mathbb{R}^n \rightarrow \mathbb{R}^{n\times n}$ be $k$-times continuously differentiable functions, with $\Sigma(x)$ always a symmetric positive-definite matrix. Fix a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in [0,\infty)},\mathbb{P})$ *(satisfying the usual conditions)* and supporting an $n$-dimensional Brownian motion $(W_t)_{t\in [0,\infty)}$. Define the $(\mathcal{F}_t)_t$-adapted process $(X_t)_{t\in [0,\infty)}$ as being the unique strong solution to:
$$
X_t = x + \int_0^t [M_s^1+M_s^2X_s] ds + \int_0^t \Sigma_s dW_s
$$

It is easy to show that *(by discrizing, using the fact that the affine transformations of Gaussians is gaussian, and that Gaussianity is preserved under the relevant limits... well summarized in this post)* under these conditions the marginals:
$$
\mu_t(B):=\mathbb{P}\left(X_t\in B \right) \qquad B\in \mathcal{F},
$$
are non-degenerate and Gaussian.

Now, let $m:[0,\infty)\to \mathbb{R}^n$ and $\sigma:[0,\infty)\rightarrow \mathbb{R}^{n\times n}$ be the functions which map a time $t$ to the respective mean and covariance of $\mu_t$. *My question is, is it true that (under mild conditions) $m$ and $\sigma$ are at-least $C^k$-functions?*

[If so, does anyone know a reference to this (likely fact)?]