Error-order conjecture for sample optimal portfolio weights

Let $\Sigma$ be the true covariance matrix, with eigenvalues $\sigma_k^2$, and let $w^*$ and $\hat{w}^*$ denote the true and sample optimal portfolio weights, respectively. For each coordinate $k$, write $\hat{w}^*_k$ and $w^*_k$ for their $k$-th entries.

Error-order conjecture. The expected coordinatewise error satisfies

$$\operatorname{\mathbb{E}}\left|\hat{w}^*_k-w^*_k\right|=O\left(\sigma_k\sqrt{\frac{p}{n}}\right),\qquad \forall\,1\leq k\leq n.$$

The constant in the order depends on the smallest and largest eigenvalues of $\Sigma$.

This conjecture summarizes the paper's experimental observations about the accuracy of sample optimal portfolio weights as the dimension-to-sample-size ratio changes. The supplied text does not establish the estimate theoretically or indicate whether it has been resolved.