Let $X,Y\in\mathbb C^{n\times n}$ be hermitian and $Y$ positive semi-definite. Does
$$
(YXY)^+=YX^+Y
$$
hold, where $(\cdot)^+$ denotes the [positive part of the respective hermitian matrix](https://planetmath.org/decompositionofselfadjointelementsinpositiveandnegativeparts) (i.e. if $A=U\operatorname{diag}(a\_1,\ldots,a\_n)U^\dagger$ hermitian with $U\in\mathbb C^{n\times n}$ unitary then $A^+=U\operatorname{diag}(\max\{a\_1,0\},\ldots,\max\{a\_n,0\})U^\dagger$)?