it's known the conjugate priors for multivariate normal distribution are the normal & inverse-whishart distributions. but i'm interest in very specific case where the correlation matrix is $R$ multiply by random variable $\kappa^2$ with inverse-gamma prior summed with $\sigma^2 I$ . I'm trying to get the closed-form posterior from an inverse gamma prior and a likelihood based on a multivariate normal distribution expecting to get inverse gamma posterior but I don't have any success . I appreciate it if you could help me get the posterior. the model:
$$ Y | \kappa \sim \mathcal{N}_j(0,( \kappa^2 R + \sigma^2 I))$$ $$ \kappa \sim \text{InverseGamma} (A_0, B_0)$$
Where $A_0$, $B_0$, and $\sigma^2$ are known constants and $R$ is $j$ by $j$ known matrix we can even say it is rank(1) matrix . $I$ is a $j$ by $j$ identity matrix.