 Solution

Use the uncentered sample covariance matrixThen $v_{\ell }=a_{\ell }^T\Sigma a_{\ell }$ and, simultaneously for every unit vector $a_{\ell }$,Here $\Vert \Sigma {\Vert }_{\mathrm{op}}=1$ and the effective rank of a covariance matrix satisfiesThe Gaussian sample-covariance operator-norm bound therefore gives, with probability at least $1-e^{-\delta }$,Under the assumed upper bound on $\log d+1+\delta$, the second term is at most the first. Thus the stronger simultaneous estimateholds for every $\ell$. Squaring and using that the displayed ratio is at most one gives the inequality requested in the question after enlarging the universal constant $C$.