Critical Hawkes process existence and uniqueness conjecture

Let $F$ be an absolutely continuous distribution function with $F(0)=0$ and density $f$, let $\tilde{F}$ be the symmetrized version of $F$, and let $\lambda>0$. A critical Hawkes process $N$ has displacement density $f$ and average intensity $\lambda$. Critical Hawkes existence and uniqueness conjecture. Such a process exists if and only if the symmetrized displacement distribution $\tilde{F}$ is transient. In that case, the critical Hawkes equation specifies a unique, stationary, and infinitely divisible distribution. This conjecture formulates the proposed equivalence between existence and transience, together with uniqueness and distributional properties in the transient case; the paper does not establish the full assertion.