Spectral equivalence of normal-scores and Kendall rank inputs

Let $\widehat{\bm{S}}_n$ be the normal-scores covariance and let $\widehat{\bm{\Sigma}}^{\tau}$ be the sin-transformed Kendall matrix. Assume $p/n\to\gamma$. Spectral equivalence of rank inputs. The two matrices have the same limiting empirical spectral distribution, even though

\left\\|\widehat{\bm{S}}_n-\widehat{\bm{\Sigma}}^{\tau}\right\\|_{\mathrm{op}}

does not vanish. A proof would need controls beyond the operator-norm and normalized Frobenius estimates currently used, which the paper's simulations indicate are insufficient for establishing the claimed spectral equivalence.