Let $y$ denote a D-dimensional multivariate normal random variable with density $$y \sim N(\mu, \Sigma(\mu)) $$ such that the covariance $\Sigma$ is a deterministic non-linear function of $\mu$. Further assume a prior distribution for $\mu$ given by $$\mu \sim N(\rho, \Omega)$$.
I have two questions. (1) Is this prior and likelihood conjugate even with $\Sigma$ being a non-linear function of $\mu$? If the answer is "it depends" please give some insights as to what conditions on $\Sigma(\mu)$ would lead to conjugacy. (2) If the answer to any of the above is "Yes" please give the posterior $p(\mu|y)$ or $p(\mu, \Sigma|y)$. Actually, even if the answer is no, but there is an analytical closed form for the posterior I would appreciate knowing what it is.