Poisson-embedding convergence conjecture for critical Hawkes processes

Let $F$ be an absolutely continuous displacement distribution, let $\lambda>0$, and let $B\in\mathcal{B}_b(\operatorname{\mathbb{R}})$. Let $N^{(g)}$ and $\lambda^{(g)}$ be defined by the Poisson-embedding recursion, and let $\tilde{F}$ be the symmetrized version of $F$. Poisson-embedding convergence conjecture. For every such $B$, $N^{(g)}(B)$ converges almost surely to a nonnegative integer as $g\to\infty$, and these limits define a point process $N^{(\infty)}$. If $\tilde{F}$ is transient, the average intensity of $N^{(\infty)}$ equals $\lambda$; otherwise it equals $0$. Moreover, $\lambda^{(g)}$ converges almost surely pointwise to a limit $\lambda^{(\infty)}$ such that $\lambda^{(\infty)}(\cdot)$ is an $\mathcal{F}^{N^{(\infty)}}$-intensity. The claim proposes a concrete construction of critical Hawkes processes and identifies the zero-intensity outcome in the transient/non-transient dichotomy; its assertions are not proved in the supplied text.