Shift-invariant optimal estimator conjecture for distributions on the real line

Let $\mu$ be a distribution on $\mathbb{R}$, and consider estimators for the associated location family. An estimator is shift-invariant if shifting every observation by $c$ shifts the estimate by $c$. Shift-invariant optimal estimator conjecture. There is an optimal estimator for $\mu$ that is shift-invariant. The conjecture is motivated by examples in which the optimal bounds for general and shift-invariant estimators coincide; the source provides no proof in the general case.