Let $X_0$ be a trace-one positive definite matrix, i.e. $X_0>0$, $\mathrm{tr}(X_0)=1$. Let $A>0$ and consider the following iteration $$ X_{k+1} = X_k^{1/2}AX_k^{1/2},\quad k\geq 0,\quad (\star) $$ where $X_k^{1/2}$ denotes the (principal) square root of $X_k$.

**My question:** Is it true that the above iteration is trace-preserving starting *from all* $X_0$'s as above, i.e. $\mathrm{tr}(X_{k+1})=\mathrm{tr}(X_{k})=1$ for all $k\geq 0$, (if and) only if $A=I$?

**N.B.** After a discussion with Nawaf Bou-Rabee, I decided to restore the original formulation of my question and accept his answer. However, I opened a new OP where you can find the edited version, which is still open.