Eigenvalues of multivariate t-wishart and wishart matrix

Question

If $W$ is a wishart matrix with Identity Covariance and $n$ degrees of freedom. And another matrix $X= v*W*S$ where $S$ is a diagonal matrix with diagonal elements as iid inverse-$\chi^2$ distributed with $v$ degrees of freedom. The mean of this $X$ is simply $v/(v-2) nI$ that is it is just mean of inverse-$\chi^2$ times mean of wishart times $v$. Similarly the variance is $var(X)=\frac{v^2}{(v-2)^2} (\frac{v-2}{v-4}E[W^2]-E[W]^2) $. I found from simulation experiments that the eigenvalues of $W$ and $X$ are also related, that is if I divide the eigenvalues of $X$ by $\frac{v}{\sqrt{(v-2)(v-4)}}$ the eigenvalues of $X$ are approximately same as that of eigenvalues of $W$, especially the largest eigenvalue of $X$, i.e. $\lambda_1(X)\frac{\sqrt{(v-2)(v-4)}}{v} \approx \lambda_1(W)$. I can think about the relation as the factor $\frac{v}{\sqrt{(v-2)(v-4)}}$ is actually the square root of the second moment (raw) of inverse-$\chi^2$ times square root of $v^2$. Essentially I am dividing the $X$ by its second raw moment, but I am not sure why this is the case why I have to correct for this second moment to see a behavior like wishart matrix eigenvalues. I performed the simulations on multiple settings for matrix dimensions as well as for various degrees of freedom, but I still find the same result, but I am not sure how to prove it mathematically. Can someone help me understand about the correction mathematically?