Resolvent convergence under heterogeneous second-moment profiles and quadratic-form control

Let $X=(x_1,\ldots,x_n)\in\mathbb R^{p\times n}$ have independent real columns with finite second moments, not necessarily centered or identically distributed. For $z\in\mathbb C\setminus[0,\infty)$, we compare \[ G^z=\left(\frac1nXX^\top-zI_p\right)^{-1} \] with a deterministic matrix $\tilde G^z$