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We follow the comparison method via coupling in [@bourgade2016fixed]. As in [@landon2019fixed], consider the interpolation between a general sample covariance matrix and the Wishart ensemble for the initial data: for any $ \nu \in [0,1] $, let $$x_k^{(\nu)}(0)=\nu s_k(0)+(1-\nu)r_k(0),\ \ \ -N {\leqslant}k {\leqslant}N,\ k \neq 0.$$ where $ s_k(t) $ and $ r_k(t) $ satisfy the singular values dynamics , with respective initial conditions a general sample covariance matrix and the Wishart ensemble. Define the corresponding dynamics of $ x_k^{(\nu)} $ to be $$\label{e.DBM}
{\mathrm{d}}x_k^{(\nu)} = \dfrac{{\mathrm{d}}B_k}{\sqrt{N}} + \left[ -\dfrac{1}{2\xi}x_k^{(\nu)} + \dfrac{1}{2}\left( \dfrac{1}{\xi} -1 \right) \dfrac{1}{x_k^{(\nu)}} + \dfrac{1}{2N} \sum_{l \neq \pm k} \dfrac{1}{x_k^{(\nu)} - x_l^{(\nu)}} \right]{\mathrm{d}}t .$$ For this Dyson Brownian motion we consider the quantity $${\mathfrak{u}}_k^{(\nu)}(t):=e^{\frac{t}{2\xi}}\dfrac{{\mathrm{d}}}{{\mathrm{d}}\nu}x_k^{(\nu)}(t).$$ From now we set $ \nu \in (0,1) $ and omit it from the notation for simplicity. A significant property is that $ {\mathfrak{u}}_k $ satisfies a non-local parabolic differential equation $$\label{e.parabolic}
\dfrac{{\mathrm{d}}}{{\mathrm{d}}t}{\mathfrak{u}}_{k}=\dfrac{1}{2}\left( 1- \dfrac{1}{\xi} \right)\dfrac{{\mathfrak{u}}_k}{x_k^2} + \dfrac{1}{2N} \sum_{l \neq \pm k}\dfrac{{\mathfrak{u}}_l - {\mathfrak{u}}_k}{\left(x_l - x_k \right)^2}.$$ Let $ {\mathfrak{v}}_k={\mathfrak{v}}_k^{(\nu)} $ solve the same equation as $ {\mathfrak{u}}_k $ in but with initial condition $ {\mathfrak{v}}_k(0)=|{\mathfrak{u}}_k(0)|=|s_k(0)-r_k(0)| $. An important result is that this equation yields a maximum principle for $ {\mathfrak{v}}_k 