Heuristic asymptotic conjecture for the free process

Let $k\geq2$, and let $\mathcal F$ be a strictly $k$-balanced $k$-uniform hypergraph with $k$-density $\rho$. Write $v(\mathcal F)$ and $e(\mathcal F)$ for its numbers of vertices and edges, and $\operatorname{aut}(\mathcal F)$ for its number of automorphisms. Let $F(n,\mathcal F)$ denote the final number of edges in the $\mathcal F$-free process on $n$ vertices. Free-process asymptotic conjecture. For every $\varepsilon>0$, there is $n_0\geq0$ such that for every $n\geq n_0$, with probability at least $1-\varepsilon$,

$$F(n,\mathcal F)=\left(\frac{1}{k!}\left(\frac{\operatorname{aut}(\mathcal F)(v(\mathcal F)-k)}{e(\mathcal F)(e(\mathcal F)-1)k!}\right)^{\frac{1}{e(\mathcal F)-1}}\pm\varepsilon\right)(\log n)^{\frac{1}{e(\mathcal F)-1}}n^{k-\frac{v(\mathcal F)-k}{e(\mathcal F)-1}}.$$

The conjecture formalizes the Poisson-paradigm heuristic for the terminal size of the free process; the source does not report a general proof.