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Abstract
Let us consider simultaneous confidence intervals for a set of linear combinations of the components of the mean vector in a multivariate normal distribution. As a special case we consider the situation when the set of linear combinations is the set of all pairwise differences. In this case Alberton and Hochberg (1984) presented so called Extended Tukey-Kramer Conjecture. This conjecture extends in certain sense the corresponding conjecture of the conservativeness of the Tukey-Kramer confidence intervals for the pairwise differences to the situation with an unknown variance matrix. Alberton and Hochberg also presented simulation results supporting the conjecture in the case of three means. In this paper the conjecture is proved in the case of three means as a corollary of a more general result.