Yes, it’s not valid to reject a null hypothesis involving multiple tests without using a multiple testing correction. The simplest conservative approach would be to use a Bonferroni correction and check whether any of the M tests has a p-value below alpha/M. Clearly, these results are not that significant. The study suggests a couple potential effects. To credibly evaluate them, the study needs to be repeated with more mice and/or fewer tests along with appropriate statistical analysis.