 Minimax sample mean for a nonnegative normal location

For independent $X_i\sim N(\theta ,1)$ with $\theta \ge 0$, the sample mean has constant squared-error risk $1/n$ and is minimax. Smooth priors supported on $[0,L]$ whose prior Fisher information is $O(L^{-2})$ give, through the Van Trees inequality, a lower bound tending to $1/n$ for every rule's worst-case risk.