Let us be given an LTI system
$$ \frac{d}{dt} x (t) = A x(t), \;\; x(0)=x_0 \\ y(t) = Cx(t) $$
where $x_0$ is a random vector (e.g. uncertainty). Then it is known that the expectation $\mathbb E[x(t)]$ is driven by the same LTI system and that the covariance $\Sigma[x_0]$ of $x_0$ propagates to the covariance of the output $\Sigma[y(t)]$ via a Lyapunov-like equation
$$ Ce^{At} \Sigma[x_0] (Ce^{At})^{\top} = \Sigma[y(t)]. $$
Question: If $(A,C)$ is observable, can one then uniquely reconstruct $\Sigma[x_0]$ from knowledge of the time-evolution of $\Sigma[y(t)]$?