Eigenvector delocalization conjecture for random graph walks

Let $G$ be the random graph and let $\nu_1,\dots,\nu_n$ be the eigenvectors appearing in Theorem 1 of the paper, with $p$ as in that theorem. For a vector $v$, write $\|v\|_\infty$ for its largest coordinate in absolute value. Eigenvector delocalization conjecture. With probability converging to $1$ as $n\to\infty$,

$$\max_{k=2,\dots,n}\|\nu_k\|_\infty^2=o\left(\sqrt{\frac{p}{n}}\right).$$

This conjecture would provide the eigenvector delocalization estimate needed to show that the relevant eigenvector sum is negligible on the paper's scale. Its status is not resolved in the supplied text.