# 2x2 ANOVA, no main/interaction effects, post-hoc cell mean comparisons necessary/allowed?

I have a 2x2 between-subject design with unequal cell sizes. I ran an ANOVA with type I sums of squares to account for the unequal cell sizes.

Neither of the two main effects nor the interaction effects were significant (all ps > 0.1).

With an overall of 2x2 = 4 cells/groups there are obviously 6 possible pairwise comparisons, i.e. 6 individual differences between cell means that might be statistically significant. Can I conclude from the non-significant main and interaction effects that none of these pairwise comparisons are statistically significant?

As the plot suggests it would be particularly interesting to examine the difference between the two means on the left-hand side, i.e gain-absent vs. gain-present.

Would it be legitimate/necessary to run a Tukey-Kramer test (due to unequal cell sizes) to test this? If so, how do I do this in R? (Here I'm assuming that the standard TukeyHSD() function would be invalid due to unequal cell sizes)