Let $A\in M_n$ and $B\in M_m$ be complex positive semi-definite matrices such that $\text{Tr}A\leq \text{Tr}B$. Let $P\in M_n$ and $Q\in M_m$ be projection matrices with the same nullity, i.e., $\ker(P)$ and $\ker(Q)$ have the same dimension. Then, can one conclude that
$$ \text{Tr}PA \leq \text{Tr} QB \,\, ?$$
No, we cannot conclude that (not even for $m=n$ and $P=Q$). Pick $A=diag(1,0)$, $B=diag(0,1)$ and $P$ the projection onto the first coordinate, then $PA=A$ and $PB=0$, so $tr(PA)=1>0=tr(PB)$.