Simultaneous statistical inference has been a cornerstone in the statistics methodology literature because of its fundamental theory and paramount applications. The mainstream multiple testing literature has traditionally considered two frameworks: the sample size is deterministic, and the test statistics corresponding to different tests are independent. However, in many modern scientific avenues, these assumptions are often violated. There is little study that explores the multiple testing problem in a sequential framework where the test statistics corresponding to the various streams are dependent. This work fills this gap in a unified way by considering the classical means-testing problem in an equicorrelated Gaussian and sequential framework. We focus on sequential test procedures that control the type I and type II familywise error probabilities at pre-specified levels. We establish that our proposed test procedures achieve the optimal expected sample sizes under every possible signal configuration asymptotically, as the two error probabilities vanish at arbitrary rates. Towards this, we elucidate that the ratio of the expected sample size of our proposed rule and that of the classical SPRT goes to one asymptotically, thus illustrating their connection. Generalizing this, we show that our proposed procedures, with appropriately adjusted critical values, are asymptotically optimal for controlling any multiple testing error metric lying between multiples of FWER in a certain sense. This class of metrics includes FDR/FNR, pFDR/pFNR, the per-comparison and per-family error rates, and the false positive rate.