The group-mean-centered vs. not decision doesn’t matter (with regard to the fixed effect estimate for the lower-level/within predictor) so long as the group mean is included as a predictor, short of one important situation: If you plan to use random slopes. I’m working from memory here, so I’m sorry if this is not quite right: 1. A random slope just on the lower-level predictor will yield different results depending on whether that lower-level predictor is mean-centered. 2. You can make them equivalent again by also adding a random effect for the group means, especially important if not group-mean-centering the within predictor.
It took me a while to figure out what was going on with that until I started playing around with the random slopes configuration.

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