Random totally simple graph conjecture

Fix an integer $k\geq 2$. Let $G$ be a uniformly random $k$-out graph on $n$ vertices, and let $\mathrm{SCC}$ be the event that $G$ is strongly connected. A graph is totally simple if it has no nontrivial graph congruences.

Random totally simple graph conjecture. One has

$$\Pr(\text{$G$ is totally simple}\mid \mathrm{SCC})\xrightarrow[n\to\infty]{}1.$$

Moreover, for fixed $n$, the function

$$k\longmapsto\Pr(\text{$G$ is totally simple}\mid\mathrm{SCC})$$

is nondecreasing. More quantitatively, there exist constants $c_k>0$ and $\beta_k\in\mathbb R$ such that

$$\Pr(\text{$G$ is totally simple}\mid\mathrm{SCC})=1-\exp(-c_kn+\beta_k)+o(1)\qquad(n\to\infty),$$

with $c_k$ increasing as $k$ increases.

The conjecture proposes that, after conditioning on strong connectivity, random fixed-out-degree graphs are typically totally simple, and it further predicts monotonicity and an exponential rate. The source provides no resolution status.