Weak discretization error conjecture for the particle system

Let $g:\mathbb{R}\to\mathbb{R}$ be smooth with bounded derivatives, let $G^P(x)=\frac{1}{P}\sum_{j=1}^P g(x_j)$, and let $\bm X_N^P$ be a realization of the particle system with continuous-time limit $\bm X^P(T)$. Weak discretization error conjecture. The weak error with respect to the continuous-time limit satisfies

$$\left|\mathbb{E}\left[G^P\left(\bm X_N^P\right)-G^P\left(\bm X^P(T)\right)\right]\right|\lesssim N^{-1}.$$

This is suggested by the weak order of the Euler–Maruyama method for sufficiently smooth coefficients and objective functions, although the coupled Brownian motion makes the rate nontrivial and the conjecture remains unproved in the stated setting.