If $A,B \in \mathbb{R}^{n \times n}$ are p.s.d matrices with $A_{ii} \geq B_{ii}$ for all $i$, can we say that $A-B$ is p.s.d

Question

This is not a homework problem.

I know that any linear combination of positive semidefinite matrices is also positive semidefinite. On the other hand, if $A,B \in \mathbb{R}^{n \times n}$ are both positive semidefinite, then $A-B$ is not positive semidefinite in general.(from my understanding)

Now, what if for each $i=1, \dots,n$, we have $(A)_{ii} \geq (B)_{ii}$, i.e. each diagonal entry in $A$ is greater than or equal to the diagonal entry in $B$ for all $i$. Can we say that $A-B$ is positive semidefinite? Note that all matrices mentioned above are symmetric.

Any reference or hint is appreciated.