If $A$ and $B$ are two positive definite matrices such that $\|A\| \leq \|B\|$ for every unitarily invariant norm $\| \cdot \|$, and $U$ is an $n\times k$ matrix with adjoint $V$ such that $VU = I_k$, then can we establish some relation between $\operatorname{trace}(VAU)$ and $\operatorname{trace}(VBU)$?