Considering in the complex fields. Let $P$ be a nonsingular matrix, $P^* $ be its conjugate transpose, is there a relation between $P^*|D|P$ and $|P^*DP|$, where $D$ is a diagonal matrix? In particular, is it true

$P^* |D|P \ge |P^*DP|$ in the sense of Lowner order, or is there an order for eigenvalues?

Here $|A|=(A^*A)^{1/2}$, the absolute value of a complex matrix.

**Edit** As I know from Suvrit's answer, there is no relation like $P^* |D|P \ge |P^* DP|$ in the sense of Lowner order. So my question becomes, is the $i$th largest eigenvalue of
$P^* |D|P$ larger than that of $|P^*DP|$?