Recursive asymptotic-variance conjecture for stratified particle filters

For $n\geq 1$, let $V_{n+1}^M(f):=\operatorname{Var}\left(M^{-1/2}\sum_{m=1}^M f(Y_{n+1}^{M,m})\right)$, and let $g_n$, $P_n$, and $\bar\eta_n^0$ denote the step-$n$ potential, mutation kernel, and limiting unnormalised measure used by the particle filter. For $f\in\mathcal{B}_b(\mathbb{R}^d)$ define

$$f_n:=g_n\left(\bar\eta_n^0(g_n)f-\bar\eta_n^0(g_nf)\right).$$

Recursive asymptotic-variance conjecture. For each $f\in\mathcal{B}_b(\mathbb{R}^d)$,

$$\left|\operatorname{Var}\left(\sqrt{M}\frac{\bar\eta_n^{0,M}(g_nf)}{\bar\eta_n^{0,M}(g_n)}\right)-\frac{V_n^M(P_nf_n)}{(\bar\eta_n^0(g_n))^4}-\frac{\bar\eta_{n-1}^0\left(g_{n-1}\left(P_nf_n^2-(P_nf_n)^2\right)\right)}{(\bar\eta_n^0(g_n))^4\bar\eta_{n-1}^0(g_{n-1})}\right|\underset{M\to\infty}{\longrightarrow}0.$$

The formula extends the explicitly verified $n=0$ variance asymptotics recursively to later particle-filter steps. It is used to derive the asymptotic variance after mutation and resampling; the source provides no proof or resolution for $n\geq1$.