In this article the Tukey-Kramer procedure for multiple comparisons of pairwise differences of mean vectors in multivariate normal distributions is considered. A multivariate version of the Tukey-Kramer procedure is presented, and a generalized Tukey conjecture of the conservativeness of the simultaneous confidence intervals for all pairwise comparisons by this procedure is affirmatively proved in the case of three correlated mean vectors. Some properties of the multivariate Tukey-Kramer procedure are also presented, and simulation results for some selected parameters are given.