Product of positive definite matrix and matrix symmetrization

Question

Suppose $X$ and $A+A^*$ are positive definite matrices. I would like to ask if it is true that $AX+ (AX)^* = AX + XA^*$ is also positive definite?

Answer 1

This should be a counterexample: $$A=\begin{pmatrix}1&2\\ -2&1\end{pmatrix},\quad X=\begin{pmatrix}2&1\\ 1&2\end{pmatrix}$$