Derive the Marchenko–Pastur law via the Stieltjes transform

Let $X$ be $n \times d$ with i.i.d. entries of mean $0$ and variance $1$, and let $S = \frac1n X^\top X$ with $d/n \to \gamma \in (0,1]$. Derive the limiting spectral density

$$\varrho_{\mathrm{MP}}(x) = \frac{\sqrt{(\lambda_+-x)(x-\lambda_-)}}{2\pi\gamma x}, \qquad \lambda_\pm = (1\pm\sqrt\gamma)^2,$$

by: defining the Stieltjes transform, deriving the self-consistent equation it satisfies in the limit, solving it, and inverting. State clearly which step is the one requiring concentration (and which this derivation therefore assumes).