Let see if I have get your point:

Now you assume jointly follows a multivariate Gaussian distribution. If you know the covariance matrix of this, then you can solve your problem.

Let

be the covariance matrix of .

You already known the the conditional distribution of and , so and are already known. I also assume that you know the "marginal" distributions of each of them, so the diagonals are also known.

The key question here is that:

So how do we interpret this?given that d only depends on kappa through r,

If is a zero matrix, then will be independent. If this is not the case we need further assumption on the structure of covariance matrix I guess? Or do you mean is conditionally independent given ?