Suppose $A$ and $B$ are positive definite matrices of the same size. Prove that
\begin{align} \operatorname{Tr}( A^{1/2} (A^{-1/2} B A^{-1/2})^{1/2} A^{1/2}) \leq \operatorname{Tr}( (A^{1/2} B A^{1/2})^{1/2}). \end{align}
Here the matrix square root is taken to be principle square root: in other words, the principal square root is the only positive definite root of a positive definite matrix.
Let $X=A(A^{-1/2}BA^{-1/2})^{1/2}$. Then the matrix on the LHS of the inequality is equal to $A^{1/2}XA^{-1/2}$ and the matrix on the RHS is equal to $(XX^\ast)^{1/2}$. Hence the inequality is equivalent to $\operatorname{tr}(X)\le\operatorname{tr}\left((XX^\ast)^{1/2}\right)$, which, as explained in my answer to your previous question, can be easily proved by performing polar decomposition on $X$.