Theorem: Let $X$ be an $n\times n$ positive semi-definite Hermitian matrix with diagonal elements $d=[d_1\cdots d_n]'$ and eigenvalues $\lambda=[\lambda_1\cdots\lambda_n]'$, where $d_1\geq\cdots\geq d_n$ and $\lambda_1\geq\cdots\geq\lambda_n$, then we have
\begin{equation} d_n\geq\lambda_n,\\ d_{n-1}d_n\geq\lambda_{n-1}\lambda_n,\\ \vdots\\ \prod_{i=1}^nd_i\geq\prod_{i=1}^n\lambda_i. \end{equation}
As you can see this theorem only works one way. I want to understand why the other way around doesn't work, i.e.: if the above inequalities hold, why not always there exists positive semi-definite $X$ with diagonal elements $d=[d_1\cdots d_n]'$ and eigenvalues $\lambda=[\lambda_1\cdots\lambda_n]'$.
The above theorem is related to theory of product majorization. I went though some books related to theory of majorization, but couldn't find a hint. Can anyone suggest me a route to try or provide some counter-example that will help me to understand the issue? Assume $\lambda_i\geq1$ for all $i$.