Lower-bound conjecture for adaptive linear-quadratic regulator regret

Let $\widehat{\pi}$ be an arbitrary adaptive policy, and let $\mathcal{R}_n(\widehat{\pi})$ denote its regret after $n$ time points. Lower-bound conjecture. For an arbitrary adaptive policy $\widehat{\pi}$ we have

$$\liminf\limits_{n \to \infty} n^{-1/2}\mathcal{R}_{n} \left(\widehat{\pi}\right)>0.$$

The conjecture asserts a universal lower bound on the regret growth of adaptive policies in linear-quadratic regulation, suggesting that no adaptive regulator can achieve regret of order smaller than $\sqrt{n}$. The supplied text presents this as an interesting direction for future work and gives no resolution.