Frieze–Pegden homomorphism conjecture for sparse random graphs

Let $c>1$ be fixed, let $G\sim G(n,c/n)$, and let $C_{2\ell+1}$ be the cycle of length $2\ell+1$. A graph homomorphism from $G$ to $C_{2\ell+1}$ is a vertex map preserving adjacency. Frieze–Pegden conjecture. There is an integer $\ell_c$ such that, with high probability, there is no homomorphism from $G$ to $C_{2\ell+1}$ for any $\ell\geq\ell_c$. The paper states that its results solve this conjecture in the regime $c=1+\varepsilon$, by showing nonexistence for $\ell=\Theta(\varepsilon^{-3})$; the minimal such $\ell(\varepsilon)$ remains of interest.