Trace inequality of positive definite matrices and diagonal matrices

Question

Assume that $A,B$ are $\mathbb{R}^{n\times n}$ positive semi-definite matrices and $C\in\mathbb{R}^{n\times n}$ is a diagonal matrix with positive diagonal entries taking values from $[0,1]$. Is it true that

$$tr(ACBC)\leq tr(AB)$$

Answer 1

No, here's a quick counterexample. $$ A = \begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix} $$ $$ B = \begin{pmatrix} 1 & 1 \\ 1& 1 \end{pmatrix} $$ $$ C = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} $$ Then $\mathrm{Tr}[AB] = 0$ but $\mathrm{Tr}[ACBC]=1$.