Suppose positive definite matrices $V, B, D\in\mathbb{R}^{n\times n}$ are given, where $D$ only contains diagonal entries of $V$, i.e., $D=diag(V)$, and $X, G\in\mathbb{R}^{n\times 2}$. Could the following be proved:
$\frac{1}{2}tr((D^{-1}G)^TVD^{-1}G) - 2tr(X^TV(D^{-1}G)) + 2tr((D^{-1}G)^TBX)<0$
for some arbitrary $G$, or for the case $G=2(VX-BX)$. I tried to apply the steps from: Possible proof for the relation involving matrix trace