Conjecture on the limiting product of critical-set size and transition probability

Let $\mathcal{C}_n^\star$ denote the critical set of configurations, let $K_n^\star$ denote the associated quantity governing the transition from the critical set, and let $\mathbb{P}_n$ be the probability law of the random graph. Scaling conjecture for the critical-set transition product. There exists a $\kappa^\star \in (1,\infty)$ such that

$$\lim_{n\to\infty} \mathbb{P}_n\left(\left|\lvert\mathcal{C}_n^\star\rvert K_n^\star-\kappa^\star\right|>\delta\right)=0 \qquad \forall\,\delta>0.$$

This conjectures concentration of the product of the critical-set cardinality and $K_n^\star$ around a finite constant greater than one. The source gives no resolution, so the conjecture remains open.