\ \ \ 1 {\leqslant}k {\leqslant}N.$$ An important idea in this paper is the following symmetrization trick (see [@che2019universality equation (3.9)]): $$s_{-i}(t)=-s_i(t),\ \ \ B_{-i}(t)=-B_i(t),\ \ \forall t {\geqslant}0,\ 1 {\leqslant}i {\leqslant}N.$$ From now we label the indices from $ -1 $ to $ -N $ and $ 1 $ to $ N $, so that the zero index is omitted. Unless otherwise stated, this will be the convention and we will not emphasize it explicitly. After symmetrization, the dynamics turns to the following form $$\label{e.SymmetrizedDynamics}
{\mathrm{d}}s_k = \dfrac{{\mathrm{d}}B_k}{\sqrt{N}} + \left[ -\dfrac{1}{2\xi}s_k + \dfrac{1}{2}\left( \dfrac{1}{\xi} -1 \right) \dfrac{1}{s_k} + \dfrac{1}{2N} \sum_{l \neq \pm k} \dfrac{1}{s_k - s_l} \right]{\mathrm{d}}t, \ \ \ -N {\leqslant}k {\leqslant}N, k \neq 0.$$

Local law and rigidity for singular values {#s.LocalLaw}
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The local law and rigidity estimates are classical results for the eigenvalues of sample covariance matrices. In this section, for later use, we rephrase these results into the corresponding version in terms of singular values.

It is well known that the empirical measure of the eigenvalues converges to the Marchenko-Pastur distribution $$\rho_{{\mathsf{MP}}}(x) = \dfrac{1}{2\pi \xi}\sqrt{\dfrac{[(x-\lambda_-)(\lambda_+ -x)]_+}{x^2}},$$ where $$\label{e.EndPoints}
\lambda_{\pm} = (1 \pm \sqrt{\xi})^2.$$ Define the typical locations of the singular values: $$\gamma_k :=  \inf 