Asymptotic independence conjecture for fractional weights in stratified resampling

Let $t\in\mathbb{N}$, let $\psi\in\mathcal{B}_b(\mathbb{R}^{t+2})$ be continuous, and let $h\in\mathcal{B}_b(\mathbb{R}^{d(t+1)})$. For the step-$n$ particles $X_n^{M,m}$, define

$$u_n^{M,m-1}:=\left\{\frac{M\sum_{i=1}^{m-1}g_n(X_n^{M,i})}{\sum_{i=1}^M g_n(X_n^{M,i})}\right\},\qquad \widetilde g_n(x):=\frac{g_n(x)}{\bar\eta_n^0(g_n)}.$$

Let $w_n^{M,m}$ denote the normalized stratified-resampling weights. Fractional-weight convergence conjecture. The difference between the two expectations obtained by evaluating $\psi$ at $(\nu_n^{M,m-1},w_n^{M,m},\ldots,w_n^{M,m+t})$ and by averaging its first argument uniformly over $[0,1]$ converges to zero after summing over $m$ and dividing by $M$:

\begin{aligned} &\left|\frac{1}{M}\sum_{m=1}^{M-t}\mathbb{E}\left[h(X_n^{M,m},\ldots,X_n^{M,m+t})\psi(\nu_n^{M,m-1},w_n^{M,m},\ldots,w_n^{M,m+t})\right]\\ &-\frac{1}{M}\sum_{m=1}^{M-t}\mathbb{E}\left[h(X_n^{M,m},\ldots,X_n^{M,m+t})\int_0^1\psi(u,\widetilde g_n(X_n^{M,m}),\ldots,\widetilde g_n(X_n^{M,m+t}))\,du\right]\right|\longrightarrow0. \end{aligned}

This is the multi-step analogue of the fractional-part convergence used for the initial resampling step and underlies the recursive variance analysis. The source gives no proof or resolution of this convergence.