A uniform Monte Carlo rate for the extended ensemble Kalman–Bucy filter

Let $d, q \ge 1$, let $R_1 \in \mathbb{R}^{d \times d}$ and $R_2 \in \mathbb{R}^{q \times q}$ be symmetric positive definite, and let $B \in \mathbb{R}^{q \times d}$ with $B^\top R_2^{-1} B = \rho I_d$, $\rho > 0$. Let $A : \mathbb{R}^d \to \mathbb{R}^d$ be $C^2$ with bounded Jacobian, with $$\lambda := -\sup_x \lambda_{\max}\bigl(DA(x) + DA(x)^\top\bigr) > 0, \qquad \|DA(x) - DA(y)\|_{\mathrm{op}} \le \kappa |x - y|, \quad \kappa > 0.$$ Define $\lambda_S = \lambda/\rho$, $\lambda_R = \lambda/\operatorname{tr}(R_1)$, $\lambda_K = \lambda/\kappa$, $$\lambda_{R,S} = \frac{\lambda_R \sqrt{\lambda_S}}{8e} \Bigl(1 + \frac{2}{\lambda_R \lambda_S}\Bigr)^{-1}, \qquad \delta = \min\bigl\{e\lambda_{R,S}, \tfrac12 \sqrt{\lambda_S}\bigr\},$$ and assume $\min\{\lambda_K \lambda_R/4, \lambda_{R,S}, \lambda_S/4\} > 1$ and $\delta \ge 4$. Let $X_0 \sim \mathcal{N}(m_0, P_0)$ with $P_0$ positive definite and $$\operatorname{tr}(P_0)^2 \le \frac{\lambda_S}{\lambda_R} \Bigl(\frac12 + \frac{1}{\lambda_R \lambda_S}\Bigr).$$ The signal and observation are $$dX_t = A(X_t) \, dt + R_1^{1/2} \, dW_t, \qquad dY_t = BX_t \, dt + R_2^{1/2} \, dV_t, \qquad Y_0 = 0,$$ with $W, V$ independent standard Brownian motions, independent of $X_0$, and the extended Kalman–Bucy filter is $$d\widehat{m}_t = A(\widehat{m}_t) \, dt + P_tB^\top R_2^{-1}(dY_t - B\widehat{m}_t \, dt), \qquad \dot{P}_t = DA(\widehat{m}_t)P_t + P_tDA(\widehat{m}_t)^\top + R_1 - P_tB^\top R_2^{-1}BP_t,$$ started from $(m_0, P_0)$. For $N \ge 2$, the extended ensemble filter is $$d\xi_t^i = \bigl[A(m_t^N) + DA(m_t^N)(\xi_t^i - m_t^N)\bigr] dt + R_1^{1/2} \, dW_t^i + p_t^N B^\top R_2^{-1} \bigl[dY_t - B\xi_t^i \, dt + R_2^{1/2} \, dV_t^i\bigr],$$ with sample mean $m_t^N$ and sample covariance $p_t^N = \frac{1}{N-1} \sum_{i=1}^N (\xi_t^i - m_t^N)(\xi_t^i - m_t^N)^\top$. The $\xi_0^i$ are independent samples of $\mathcal{N}(m_0, P_0)$, independent of the signal and all noises, and the $W^i, V^i$ are mutually independent standard Brownian motions, independent of the signal noises and the initial values. The comparison particle starts at $\zeta_0^1 = \xi_0^1$ and solves $$d\zeta_t^1 = \bigl[A(\widehat{m}_t) + DA(\widehat{m}_t)(\zeta_t^1 - \widehat{m}_t)\bigr] dt + R_1^{1/2} \, dW_t^1 + P_tB^\top R_2^{-1} \bigl[dY_t - B\zeta_t^1 \, dt + R_2^{1/2} \, dV_t^1\bigr].$$ Write $W_r$ for the $r$-Wasserstein distance, with the norm $\|(m, P)\|^2 = |m|^2 + \|P\|_F^2$ on mean–matrix pairs. Under these hypotheses Del Moral, Kurtzmann and Tugaut proved the estimate below with $N^{-\beta}$ in place of $N^{-1/2}$, for some $N_0 \ge 1$ and some $\beta \in (0, 1/2]$, and they conjecture that the optimal rate is $\beta = 1/2$, as in the linear-Gaussian case.

Problem. Are there finite constants $C, N_0$, depending on the fixed model and initial law but not on $N$ or $t$, such that every integer $N \ge N_0$ satisfies $$\sup_{t \ge 0} \max\Bigl\{ W_\delta\bigl(\operatorname{Law}(m_t^N, p_t^N), \operatorname{Law}(\widehat{m}_t, P_t)\bigr), \; W_2\bigl(\operatorname{Law}(\xi_t^1), \operatorname{Law}(\zeta_t^1)\bigr) \Bigr\} \le C N^{-1/2}?$$ All laws are unconditional, averaging over the observations as well as the ensemble.