Is it in general true that $|X+Y|\leq|X|+|Y|$ (in the sense that $RHS-LHS$ is positive-semidefinite) for any complex square matrices $X,Y$, where $|X|\colon=\sqrt{X^{*}X}$? If yes, is it still true for infinite-dimensional Hilbert spaces? If no, what is a counterexample, and is there a known nontrivial condition which rules out any counterexample?
I believe the inequality is not true even for $2\times 2$ matrices, but not sure about counterexamples.