Conjecture on the extensive metastable energy barrier

Let $\Gamma_N^\star$ denote the metastable energy barrier for the dynamics on the random graph with $N$ vertices, and let $\mathbb{P}_N$ be the corresponding probability law. Conjecture on the metastable energy barrier. There exists a $\gamma^\star \in (0,\infty)$ such that

$$\lim_{N\to\infty} \mathbb{P}_N\Big( \big| N^{-1} \Gamma^\star_N - \gamma^\star\big| > \delta\Big) = 0 \qquad \forall\,\delta>0.$$

This conjecture predicts that the metastable energy barrier is extensive and obeys a law of large numbers. The source presents it as one of several conjectures from Dommers, den Hollander, Jovanovski, and Nardi; no resolution is given.