For a two-tailed Bonferroni pairwise test comparing groups X and Y, which is the correct null hypothesis to test?

If you plan to perform m = 10 pairwise tests and want to keep familywise alpha = 0.05, what Bonferroni-adjusted per-comparison alpha should you use? If a particular unadjusted p = 0.004, does it survive the correction?

A researcher analyzes the mean scores of four different teaching methods using ANOVA. She then performs both the Bonferroni and Tukey post hoc tests for all pairwise comparisons. The Bonferroni test finds that only the comparison between method A and method D is significant at the $0.05$ level. The Tukey test finds that both method A vs. method D and method B vs. method D are significant at the $0.05$ level. Based on these results, which statement is most accurate?