Nonvanishing conjecture for eigenfunctions of random graphs

Let $p\in(0,1)$ be fixed, let $G\sim G(n,p)$, and let $f$ be an eigenfunction of $G$. The zero set of $f$ is

$$\{v\in V(G):f(v)=0\}.$$

Nonvanishing conjecture. Almost surely, every eigenfunction $f$ of $G$ has empty zero set:

$$\{v\in V(G):f(v)=0\}=\emptyset.$$

This asks whether eigenvectors of a dense Erdős–Rényi random graph almost surely have no zero coordinates. It is posed as a future direction in the source, and no resolution is supplied there.