Essential boundedness conjecture for geometric ergodicity of i-cSMC

Let $T\in\mathbb{N}$, let $\pi_t$ denote the time-$t$ marginal of the target distribution, and let $G_t$ be the corresponding potential function. Write $P_N$ for the iterated conditional SMC (i-cSMC) kernel with $N$ particles. For a measurable function, $\pi_t$-${\rm ess}\sup_{x_t}G_t(x_t)$ denotes its essential supremum with respect to $\pi_t$. Essential boundedness conjecture. If

$$\pi_t\text{-}{\rm ess}\sup_{x_t}G_t(x_t)=\infty$$

for some $t\in[T]$, then the i-cSMC kernel is not geometrically ergodic for any $N\in\mathbb{N}$. This conjecture proposes necessity of essential boundedness of the potentials for geometric ergodicity. The paper notes that this conclusion holds in several examples through sticky sets, but does not establish the general claim; hence its status remains open.