# Correction for multiple comparisons in mixed-effects model?

I'm running a logistic mixed-effects model in which I predict word knowledge from condition (three levels: experimental1, experimental2, control), time of testing (immediately, delayed), and the interaction between condition and testing moment.

The model output looks like this:

Fixed effects            Estimate (logit)   Std. error  z-value p-value

(Intercept)             -2.32               0.40       -5.78    < .001
Condition:Experimental1  0.96               0.30        3.16    .002
Condition:Experimental2  0.78               0.31        2.52    .01
Testing moment:Delayed  -0.04               0.10       -0.46    .65
Cond:Exp1 * Test:Delay  -0.09               0.13       -0.72    .47
Cond:Exp2 * Test:Delay  -0.16               0.14       -1.22    .22

Random effects           Variance   Std. dev.

Participant (intercept)  0.93       0.97
Word (intercept)         1.85       1.36


I can see that the contrast Experimental1 - Control has a p-value of .002, and the contrast Experimental2 - Control has a p-value of .01. Through relevelling, I found that the p-value for the contrast Experimental1 - Experimental 2 is .56.

My question is whether any correction for multiple testing should be applied. For the factor of condition, should alpha be .05 / 3 = .0167?

If so, does alpha stay at .05 for Testing moment, where only one comparison can be made? (immediate - delayed)