For $X\in\mathbb{R}^{n\times p}$ with $\operatorname{rank}(X)=p$, and positive definite $V\in\mathbb{R}^{n\times n}$, consider the following matrix $$A=X^+V(X^+)^T-(X^TV^{-1}X)^{-1}$$ Where $X^+$ denotes the Moore-Penrose pseudoinverse of $X$, i.e. $(X^TX)^{-1}X^T$.
I know that both terms are positive definite and that $A=0$ when $X$ is square, but I'm not sure how to show that $A$ is positive semi-definite (if it even is).