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List et al. (2019) provides a framework for testing multiple null hypotheses simultaneously using experimental data in which simple random sampling is used to assign treatment status to units. As in List et al. (2019), we rely on general results in Romano and Wolf (2010) to develop under weak assumptions a procedure that (i) asymptotically controls the familywise error rate - the probability of one or more false rejections - and (ii) is asymptotically balanced in that the marginal probability of rejecting any true null hypothesis is approximately equal in large samples. Our analysis departs from List et al. (2019) in that it further exploits observed, baseline covariates. The precise way in which these covariates are incorporated is based upon results in Lin (2013) in order to ensure that inferences are typically more powerful in large samples.