Conjecture on eigenvector coordinate averages for large S-regular matrices

Let $T$ be a large $S$-regular matrix, and let $T^\ast$ be the corresponding $S$-regular matrix on $\operatorname{\mathcal{T}}_S$. Let $U\subset V_i$ be a sufficiently large subset of the $i$-th partition cell of the underlying graph of $T$. Let $\phi$ be a normalized eigenvector of $T$ with eigenvalue $\lambda$, and let $\mu_i$ be defined as in the paper's vertex-distribution proposition. Write $c_i=|V_i|/|V|$.

Eigenvector coordinate conjecture. With high probability,

$$\sum_{v\in U}\phi(u)\approx 0,$$

and

$$\frac{|V|}{|U|}\sum_{v\in U}\left(\phi(u)\right)^2\approx\frac{\mu_i(\lambda)}{\sum_j^k c_j\mu_j(\lambda)}.$$

Here the conjecture predicts that most eigenvectors have approximately zero mean on a large subset of a partition cell, while their average squared coordinates are governed by the limiting spectral measures. The preceding results provide the limiting cumulative distributions, but the asserted coordinate-level approximations remain conjectural.