I try to get a sense of the correct adjusted Holm-Bonferroni significance levels $\alpha$ when dealing with multiple interaction terms. To simplify, assume one has the following model: $$ y=\beta_0 + \beta_1x_1 + \beta_2x_2+\beta_3x_3+\beta_4x_1x_3+\beta_5x_2x_3 + \varepsilon $$
I would publish any of the following statistical findings (Theories T):
T1a: $\beta_1\ne0$ and $\beta_4=0$
T1b: $\beta_1\ne0$ and $\beta_4\ne0$
T2a: $\beta_2\ne0$ and $\beta_5=0$
T2b: $\beta_2\ne0$ and $\beta_5\ne0$
The crucial point is that I would not publish $\beta_1=0$ and $\beta_4\ne0$ and would also not publish $\beta_2=0$ and $\beta_5\ne0$. Thus, I test $\beta_4$ and $\beta_5$ only conditional on $\beta_1$ and $\beta_2$, respectively, being significant.
Let's assume now, we have the following p-values: 0.01 for $\beta_1$, 0.02 for $\beta_2$, 0.04 for $\beta_4$ and 0.05 for $\beta_5$. We test for the overall significance level $\alpha=0.05$.
Using the Holm-Bonferroni approach, I would assume that the significance level for $\beta_1$ is $\alpha/2=0.25$. That is because the interaction terms $\beta_4$ and $\beta_5$ are never tested before $\beta_1$ and $\beta_2$, respectively, are significant. For $\beta_2$, I would put the significance level at $\alpha/2=0.25$. That is because I am not testing $\beta_5$ before $\beta_2$ and we already dealt with $\beta_1$. That means at this point, I only considered $\beta_2$ and $\beta_4$. For $\beta_4$, I would put the significance level at $\alpha/2=0.25$. That is because I already dealt with $\beta_1$ and $\beta_2$ - that is now identical to the standard Holm-Bonferroni test. Lastly for $\beta_5$, the significance level is $\alpha/1=0.05$ as usual.
Thus in this example, I would accept T1 and T3 and reject T3 and T4 using the Holm-Bonferroni correction. Is my dealing with the interaction effects $\beta_4$ and $\beta_5$ correct?
NOTE 1: In my actual problem, I have two separate models $y=\beta_0 + \beta_1x_1 +\beta_3x_3+\beta_4x_1x_3 + \varepsilon $ and $y=\beta_0 + \beta_2x_2+\beta_3x_3+\beta_5x_2x_3 + \varepsilon $ fitted with two different datasets. But I don't think this makes any difference in this case.
NOTE 2: The question How to apply Bonferroni correction when including an interaction term? has some similarities to this one, but I am not sure whether the answers also apply to my question.