On the procedure of hodges and lehmann for testing composite hypotheses of the mean of a normal distribution with unknown variance

In 1954 Hodges and Lehmann gave a test procedure for testing the hypothesis that the mean of an identically independently normally distributed random sample with unknown variance is contained within a certain interval μ1, μ2]. The test is similar on the boundary of the zero-hypothesis and superior in power to the composite t-test usually applied to this problem. However Hodges and Lehmann could prove the unbiasedness of their test only for the special case that the sample consists of two elements. From numerical computations they guessed that unbiasedness would be valid for arbitrary sample sizes. This question is discussed here and partially answered.