  In this paper, we extend results of Eigenvector Thermalization to the case of
generalized Wigner matrices. Analytically, the central quantity of interest
here are multiresolvent traces, such as $\Lambda_A:= \frac{1}{N} \text{Tr }{
GAGA}$. In the case of Wigner matrices, as in
\cite{cipolloni-erdos-schroder-2021}, one can form a self-consistent equation
for a single $\Lambda_A$. There are multiple difficulties extending this logic
to the case of general covariances. The correlation structure prevents us from
deriving a self-consistent equation for a single matrix $A$; this is due to the
introduction of new terms that are quite distinct from the form of $\Lambda_A$.
We find a way around this by carefully splitting these new terms and writing
them as sums of $\Lambda_B$, for matrices $B$ obtained by modifying $A$ using
the covariance matrix. The result is a system of self-consistent equations
relating families of deterministic matrices. Our main effort in this work is to
derive and analyze this system of self-consistent equations.
