Variance-decay conjecture for the particle estimator

Let $\mathcal{A}(M,P,N)$ be the single-level estimator, and let $\bm X_N^P$ denote the particle-system realization. For $M$ samples and $P$ particles, variance-decay conjecture. The estimator variance satisfies

$$\operatorname{Var}\left[G^P\left(\bm X_N^P\right)\right]\lesssim P^{-2},$$

corresponding to variance of order $\mathcal{O}(M^{-1}P^{-2})$ for the sample average. The conjecture is motivated by observed strong convergence of order $P^{-1}$ rather than the standard order $P^{-1/2}$ and by a variance bound in a simplified setting.