Following up on the negative answer to this question, I would be interested in knowing the answer to the following question, which I cannot seem to find an obvious contradiction to when testing for simple cases, such as the hypercube and the circle:
Let $A$ be the adjacency matrix of a vertex transitive graph $G$ with eigenvalues $\lambda_n \leq \lambda_{n-1} \leq \ldots \leq \lambda_2 < \lambda_1 = 1$. Given the eigenspace $\Psi_2$ of $A$ corresponding to the eigenvalue $\lambda_2$, can we always find an eigenvector $\psi \in \Psi_2$ with $\psi \cdot \psi = |V(G)|$ and $\|\psi\|_\infty \leq C$, for some universal constant $C$ which do not depend on the chosen graph?
The motivation for that something like this could be true is that it feels like, and I realize that this sounds very vague, that for one single entry to be large, and for there to be many symmetries in the graph, interpolating between $ \psi $ and $\psi \circ \varphi$ for some $\varphi \in Aut(G)$ and renormalizing should yield a more "smooth" vector, i.e. contradicting that the initial entry was so large.