A non-asymptotic bound on the TV distance between a Wishart matrix and an appropriately scaled GOE matrix

In this note, we prove a non-asymptotic version of a theorem by Rácz and Richey, showing that a Wishart matrix is close in total variation to an affine transformation of a GOE matrix. The proof mirrors a proof in a paper by Bubeck, Ding, Eldan, and Rácz, with some changes made to make it non-asymptotic.