Adaptive optimality of data-driven concentration coordinates

Let $\mu$ be a distribution and let $\mathcal{F}[\mu,\psi]$ denote the relevant concentration objective for a coordinate $\psi$. Given $n$ samples, let $\hat{\psi}_n$ be a data-driven procedure for selecting a coordinate. Adaptive concentration conjecture. There exists such a procedure satisfying

$$\mathcal{F}[\mu,\hat{\psi}_n]\leq\inf_\psi\mathcal{F}[\mu,\psi]+O(n^{-1/2}).$$

This conjecture asks whether the optimal concentration coordinate can be learned from data at a parametric rate. The supplied source gives no evidence establishing or refuting the claim.