Below I produce plots of correlation as a function of time separation. We can talk about this. I agree that it is more flexible than AR1, but do not agree that it can completely take on the function form of an AR1 structure. Is induces correlations that necessarily change with time (to the extent that the random slope variance component is big). So, the correlation between observations 2 days apart is different early in a study as compared to later in the study. Of course the variance also changes with time.
I think that if you put random intercept and slopes in the model you will often not need additional AR1 correlation but not always.
You should plot person specific trajectories. I think it is likely that random intercept and slope could do a good job in the violet2 data.
