Conjecture on the scaling of the critical height

Let $Gamma_n^\star$ denote the energy barrier and let $\mathbb{P}_n$ be the probability law of the random graph. Scaling conjecture for the critical height. There \exists a $\gamma^\star \in (0,\infty)$ such that

$$\lim_{n\to\infty} \mathbb{P}_n\left(\left|n^{-1}\Gamma_n^\star-\gamma^\star\right|>\delta\right)=0 \qquad \forall\,\delta>0.$$

This conjectures convergence in probability of the critical energy barrier per vertex to a positive constant, describing the extensive scaling of the metastable barrier. The source gives no resolution, so the conjecture remains open.