I have found two different expressions for the Jeffreys prior of a multivariate Gaussian. Eq. (3) in this article states that $$p(\mu,\Sigma) \propto \det(\Sigma)^{-(d+2)/2}$$
However in page 73 of this book it is claimed that $$p(\mu,\Sigma) \propto \det(\Sigma)^{-(d+1)/2}$$
Furthermore, in the first article it says that the proposed density
[...] is the only density which makes the Fisher information for the parameters invariant to all possible reparameterizations of the Gaussian.
Wouldn't this mean that the second prior cannot be Jeffreys, by definition? Is the second one wrong then?