What are the parameters of a Wishart-Wishart posterior? When infering the precision matrix $\boldsymbol{\Lambda}$ of a normal distribution used to generate $N$ D-dimensional vectors $\mathbf{x_1},..,\mathbf{x_N}$
\begin{align}
\mathbf{x_i} &\sim \mathcal{N}(\boldsymbol{\mu, \Lambda^{-1}}) \\
\end{align}
we usually place a Wishart prior over $\boldsymbol{\Lambda}$ since the Wishart distribution is the conjugate prior for the precission of a multivariate normal distribution with known mean and unknown variance:
\begin{align}
\mathbf{\Lambda} &\sim \mathcal{W}(\upsilon, \boldsymbol{\Lambda_0}) \\
\end{align}
where $\upsilon$ are the degrees of freedom and $\boldsymbol{\Lambda_0}$ the scale matrix. To add robustness and flexibility to the model we put an hyperprior over the parameters of the Wishart. For instance, Görür and Rasmussen suggest:
\begin{align}
\mathbf{\Lambda_0} &\sim \mathcal{W}(D, \frac{1}{D}\boldsymbol{\Lambda_x}) \\
\frac{1}{\upsilon-D + 1} &\sim \mathcal{G}(1, \frac{1}{D}) \\
\end{align}
where $\mathcal{G}$ is tha Gamma distribution.
Question:
in order to sample the posterior of $\boldsymbol{\Lambda_0}$ 
\begin{align}
p(\boldsymbol{\Lambda_0 | X, \Lambda}, \upsilon, D, \boldsymbol{\Lambda_x}) \propto
\mathcal{W}(\boldsymbol{\Lambda} | \upsilon, \boldsymbol{\Lambda_0})
\mathcal{W}(\boldsymbol{\Lambda_0} |D, \frac{1}{D}\boldsymbol{\Lambda_x}) \\  
\end{align}
what is the family and parameters of this posterior?
PS:
Dropping all factors that do not depend on $\boldsymbol{\Lambda_0}$ and identifying the parameters with the parameters of a Wihsart I get a Wishart with parameters:
\begin{align}
\upsilon' &= \upsilon + D\\
\boldsymbol{\Lambda'} &= \boldsymbol{\Lambda} + \boldsymbol{\Lambda_x}
\end{align}
which looks quite nice, but I am not confident at all since I don't find any example neither on books nor the internet.
Erratum:
Görur and Rasmussen suggest those hyperpriors over the Wishart parameters, but this equation:
\begin{align}
\mathbf{\Lambda} &\sim \mathcal{W}(\upsilon, \boldsymbol{\Lambda_0}) \\
\end{align}
should be instead:
\begin{align}
\mathbf{\Lambda} &\sim \mathcal{W}(\upsilon, \boldsymbol{\Lambda_0}^{-1}) \\
\end{align}
therefore solving the lack of conjugacy. If we want to keep $\boldsymbol{\Lambda_0}$ then we should use the Inverse Wishart as a prior (see @Xi'an 's answer)
 A: Ok, thanks to @Xi'an answer I could make the whole derivation. I will write it for a general case:
\begin{align}
\mathcal{W}(\mathbf{W} | \upsilon, \mathbf{S^{-1}} ) \times \mathcal{W}(\mathbf{S} | \upsilon_0, \mathbf{S_0})
\end{align}
where the $\mathbf{S^{-1}}$ is the key to conjugacy. If we want to use $\mathbf{S}$ then it should be :
\begin{align}
\mathcal{W}(\mathbf{W} | \upsilon, \mathbf{S} ) \times \mathcal{IW}(\mathbf{S} | \upsilon_0, \mathbf{S_0})
\end{align}
I'm doing the first case (please correct me if I am wrong):
\begin{align}
\mathcal{W}(\mathbf{W} | \upsilon, \mathbf{S^{-1}} ) \times \mathcal{W}(\mathbf{S} | \upsilon_0, \mathbf{S_0})
&\propto 
|\mathbf{S}|^{\upsilon/2} \exp\{-\frac{1}{2} tr(\mathbf{SW}) \}\\
&\times 
|\mathbf{S}|^{\frac{\upsilon_0 - D -1 }{2}} \exp\{-\frac{1}{2} tr (\mathbf{S_0^{-1} S})\}\\
&\propto 
|\mathbf{S}|^{\frac{\upsilon + \upsilon_0 - D -1 }{2}} \exp\{-\frac{1}{2} tr ( (\mathbf{W} + \mathbf{S_0^{-1}) S})\}
\end{align}
where we used the fact that $tr({\mathbf{SW}}) = tr({\mathbf{WS}})$. By inspection, we see that this is a Wishart distribution:
\begin{align}
p(\mathbf{S} | \cdot) = \mathcal{W}(\upsilon+ \upsilon_0, \mathbf{(W+S_0^{-1})^{-1}})
\end{align}
Extension for $N$ draws $\mathbf{W_1...W_N}$:
For the case when we have $N$ precision matrices then the likelihood becomes a product of $N$ likelihoods and we get:
\begin{align}
p(\mathbf{S} | \cdot) = \mathcal{W}(N \upsilon+ \upsilon_0, (\sum_{i=1}^N \mathbf{W_i+S_0^{-1})^{-1}})
\end{align}
A: The product of the two densities in
$$
p(\boldsymbol{\Lambda_0 | X, \Lambda}, \upsilon, D, \boldsymbol{\Lambda_x}) \propto
\mathcal{W}(\boldsymbol{\Lambda} | \upsilon, \boldsymbol{\Lambda_0})
\mathcal{W}(\boldsymbol{\Lambda_0} |D, \frac{1}{D}\boldsymbol{\Lambda_x}) \\  
$$
leads to
\begin{align*}
p(\boldsymbol{\Lambda_0 | X, \Lambda}, \upsilon, D, \boldsymbol{\Lambda_x}) &\propto
|\boldsymbol{\Lambda_0}|^{-\upsilon/2}\,\exp\{-\text{tr}(\boldsymbol{\Lambda_0}^{-1}\boldsymbol{\Lambda})/2\}\\ &\times |\boldsymbol{\Lambda_0}|^{(D-p-1)/2}\,\exp\{-D\,\text{tr}(\boldsymbol{\Lambda_x}^{-1}\boldsymbol{\Lambda_0})/2\}\\
&\propto|\boldsymbol{\Lambda_0}|^{(D-\upsilon-p-1)/2}\,\exp\{-tr(\boldsymbol{\Lambda_0}^{-1}\boldsymbol{\Lambda}+D\,\boldsymbol{\Lambda_x}^{-1}\boldsymbol{\Lambda_0})/2\}\,,
\end{align*}
which does not appear to be a standard density. To keep conjugacy of sorts, the right hierarchical prior on $\boldsymbol{\Lambda_0}$ should be something like
$$
\boldsymbol{\Lambda_0}\sim\mathcal{IW}(\boldsymbol{\Lambda_0} |D, \frac{1}{D}\boldsymbol{\Lambda_x})\,.
$$
