Ergodicity conjecture for the inhibited discrete-time Hawkes process

Let $a,b,c\in\mathbb{R}$ be parameters of the Markov chain $(X_n)$. Let $P$ be the polynomial associated with these parameters, and write $\operatorname{Disc}(P)$ for its discriminant. Ergodicity conjecture. If $b\leq 1$ and $c<0$ are such that

$$\operatorname{Disc}(P)<0,$$

then $(X_n)$ is an ergodic Markov chain.

This conjecture proposes a slight relaxation of the assumptions used in the paper's proven recurrence theorem, motivated by numerical experiments. The available context does not establish the claim, so its resolution remains open.