Can fixed-effects become biased due to random structure misspecification I'm following-up on this great answer. Essentially, I was wondering how could misspecification of random-effects bias the estimates of fixed-effects?
So, can the same set of fixed-effect coefficients become biased if we create models that only differ in their random-effect specification?
Also as a conceptual matter, can we say in mixed-effect models, the fixed-effect coef is some kind of (weighted) average of the individual regression counterparts fit to each individual cluster and that is why fixed-effect coefs in mixed models can prevent something like this Simpson's Paradox case from happening?
A possible R demonstration is appreciated.

This paper may (or may not!) be of interest
https://www.sciencedirect.com/science/article/abs/pii/S0378375820300732?via%3Dihub

We explore how violations of the often-overlooked standard assumption
that the random effects model matrix in a linear mixed model is fixed
(and thus independent of the random effects vector) can lead to bias
in estimators of estimable functions of the fixed effects. However, if
the random effects of the original mixed model are instead also
treated as fixed effects, or if the fixed and random effects model
matrices are orthogonal with respect to the inverse of the error
covariance matrix (with probability one), or if the random effects and
the corresponding model matrix are independent, then these estimators
are unbiased. The bias in the general case is quantified and compared
to a randomized permutation distribution of the predicted random
effects, producing an informative summary graphic for each estimator
of interest. This is demonstrated through the examination of sporting
outcomes used to estimate a home field advantage.

