Let $A, B\in M_{n}(\mathbb{R})$ be symmetric positive definite matrices. It is easy to see $Tr(A^2+AB^2A)=Tr(A^2+BA^2B)$. Numerical experiments indicate $$Tr[(A^2+AB^2A)^{-1}]\ge Tr[(A^2+BA^2B)^{-1}],~~(1)$$ but it seems difficult to show it.

Remark. When $n=2,3$, by direct computation, (1) is true. Here is an expriment done by matlab:

for s=1:1000

```
as=randn(4,4);
bs=randn(4,4);
ts=as*as';
rs=bs*bs';
ls=trace(inv(ts^2+ts*rs^2*ts)-inv(ts^2+rs*ts^2*rs))
```

end

{\bf Updated.} What about $A, B\in M_{n}(\mathbb{C})$ be positive definite Hermitian matrices.