A condition on the sampling design which previously  was shown to be a sufficient condition for the Horvitz-Thompson estimator to be the unique hyperadmissible estimator of the population total is now shown to be a necessary condition also save for trivial exceptions. The exceptions are the sampling designs in which the only samples with positive probability are such that they include the whole population.
V. M. Joshi. "A Note on Hyperadmissibility of Estimators for Finite Populations." Ann. Math. Statist. 43 (4) 1323 - 1328, August, 1972. https://doi.org/10.1214/aoms/1177692484