It is shown that, for sequences of median unbiased estimates, the asymptotic efficiency never exceeds Fisher's bound. In other words: assumptions on the convergence of the distributions of the estimates are superfluous in this case.
J. Pfanzagl. "On the Asymptotic Efficiency of Median Unbiased Estimates." Ann. Math. Statist. 41 (5) 1500 - 1509, October, 1970. https://doi.org/10.1214/aoms/1177696795