Consider the linear regression model
$$b = Xy + e, \quad E[e] = 0, \quad E[ee'] = V$$
Assume that the matrix $X$ has linearly independent columns. It is well known that the minimum variance affine unbiased estimator of $y$ is
$$\hat{y} = (X'WX)^+ X'Wb $$
where superscript + denotes Moore-Penrose inverse and $W$ is the optimal weighting matrix is
$$W = (V + XTX')^+$$
where $T$ may be any positive semidefinite matrix such that
$$\text{col}\, X \subseteq \text{col}\, (V + XTX')$$
I am trying to directly verify that the quantity $(X'WX)^+X'W$ is independent of the choice of $T$, so long as $T$ meets the above conditions. That is, I want to show
$$[X'(V + XX')^+X]^+X'(V + XX')^+ = [X'(V + XTX')^+X]^+X'(V + XTX')^+$$
I have been unable to arrive at a proof for this. Can anyone help prove the result, or provide a counterexample?