Coefficients for all levels of a categorical factor unchanged in lmer after adding variables

Question

I was sent over here from stack overflow here: https://stackoverflow.com/questions/62635123/coefficients-for-all-levels-of-a-categorical-factor-unchanged-in-lmer-after-addi (sorry for posting there first / now double posting, I am still kind of new here).

I am estimating multilevel models with lmer with subject random effects for a within-subject design study. I have a measurement of a dependent variable for each subject in three different treatment conditions, resulting in a balanced design. In addition to the treatment dummies I also have control variables in the lmer model.

First thing that stuck out is that all treatment dummies had equal standard errors, which has already been asked and answered here:

Equal standard errors for all levels of a categorical factor in lmer

The second thing that stuck out was that the coefficients of the treatment dummies do not change if I add control variables to the model.

Here the behavior of lmer is reproduced with some simulated data:

library(tidyverse)
library(lme4)
library(lmerTest)

#Some data:
id <- rep(1:50) #subject id
dependent_1 <- rnorm(50,10,5) #dependent measure in treatment 1
dependent_2 <- rnorm(50,18,3) #dependent measure in treatment 2
dependent_3 <- rnorm(50,28,4) #dependent measure in treatment 3
control_a <- rnorm(50, 100, 5) #first control
control_b <- rnorm(50, 200,33) #second control

df <- data.frame(id, dependent_1, dependent_2, dependent_3, control_a, control_b) #make dataframe

#Reshape to long form
df_long <- pivot_longer(df, 
                        cols = starts_with("dependent_"), 
                        names_to = c(".value","treatment"),
                        names_sep = "\\_")

#Treatment to factor
df_long$treatment <- as.factor(df_long$treatment)

#LMER Models
lmer_model.1 <- lmer(dependent ~ treatment +(1|id), data = df_long, REML = FALSE) #Model with treatment dummies only
lmer_model.2 <- lmer(dependent ~ treatment + control_a + control_b + (1|id), data = df_long, REML = FALSE) #Model with treatment dummies and controls

I get the following results:

===============================================================
                            Model 1               Model 2             
---------------------------------------------------------------
(Intercept)             9.246 (0.567) ***    17.535 (7.796) *  
treatment2              8.157 (0.787) ***     8.157 (0.787) ***
treatment3             20.030 (0.787) ***    20.030 (0.787) ***
control_a                                    -0.067 (0.072)    
control_b                                    -0.008 (0.011)    
---------------------------------------------------------------
AIC                   852.194               854.977            
BIC                   867.247               876.051            
Log Likelihood       -421.097              -420.488            
Num. obs.             150                   150                
Num. groups: id        50                    50                
Var: id (Intercept)     0.596                 0.457            
Var: Residual          15.492                15.492            
===============================================================
*** p < 0.001; ** p < 0.01; * p < 0.05

In the old question at stackoverflow this was attributed to the controls having a correlation of zero with the treatment dummies (which makes sense, as to the way the example data is set up and if the predictors are not correlated the estimations should not change when they are added to model stepwise). However, I would think that is also the case for any within-subject design? So this should always happen in whithin-subject designs with treatment dummies and subject characteristics as controls?