the notation $ A \lesssim B $ if there exists a constant $ C $ which is independent of $ N $ such that $ A {\leqslant}CB $ holds. We also denote $ A \sim B $ if both $ A \lesssim B $ and $ B \lesssim A $ hold. If $ A $ and $ B $ are complex valued, $ A \sim B $ means $ {\text{\rm Re}\hspace{0.1cm}}A \sim {\text{\rm Re}\hspace{0.1cm}}B $ and $ {\text{\rm Im}\hspace{0.1cm}}A \sim {\text{\rm Im}\hspace{0.1cm}}B $. We also denote $ C $ a generic constant which does not depend on $ N $ but may vary form line to line. We use $ \llbracket A,B \rrbracket := [A,B] \cap {\mathbb{Z}}$ to denote the set of integers between $ A $ and $ B $.

We also denote $$\varphi = e^{C_0 (\log \log N)^2}$$ a subpolynomial error parameter, for some fixed $ C_0>0 $. This constant $ C_0 $ is chosen large enough so that the eigenvalues (and singular values) rigidity and the strong local Marchenko-Pastur law hold (see section \[s.LocalLaw\]).

Acknowledgement {#acknowledgement .unnumbered}
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The author would like to thank Prof. Paul Bourgade for suggesting this problem, helpful discussions, and useful comments on 