For Hermitian matrices $X, Y$, I write $X\ge Y\ge 0$ to mean $X-Y$ and $Y$ are positive semidefinite.
In Lemma 2.5 of [Linear Algebra Appl. 452 (2014) 1-6] I proved that if $X + Y\ge W + Z$, $X\ge W\ge Y\ge 0$ and $X\ge Z \ge Y\ge 0$, then $$\det X+\det Y\ge \det W+\det Z.$$
I was thinking of relaxing the condition a little bit.
Is the same inequality true under only the condition $X + Y\ge W + Z$, $X\ge Y\ge 0$, $X\ge W\ge 0$, $X\ge Z\ge 0$?