Groeneboom's local limit conjecture for decreasing deconvolution mixtures

Let $g$ be a right-continuous decreasing density on $[0,\infty)$ with finitely many discontinuity points $a_0=0<a_1<\dots<a_m$, and suppose that $g'$ satisfies

$$\int_{(0,\infty)}\frac{g'(x)^2}{g(x)}\,dx<\infty.$$

Assume the integrand is zero at the discontinuity points and where $g$ is zero; assume also that $g'$ is bounded and continuous on each $(a_{i-1},a_i)$, with $a_{m+1}=\infty$, and that there are positive constants $k_1,k_2$ such that

$$|g'(t+u)|\leq k_1|g'(t)|$$

whenever $0<u<k_2$ and $a_i<t<t+u<a_{i+1}$ for some $0\leq i\leq m$. Let $F_0$ be the distribution function of nonnegative random variables $X_i$, continuously differentiable at $z_0>0$ with $f_0(z_0)>0$, and let the convolution density be

$$h(z)=\int g(z-x)\,dF_0(x),\qquad z\geq 0.$$

Groeneboom's deconvolution local limit conjecture. For the estimator $F_n^{(1)}$ appearing in the decreasing-mixture deconvolution model,

$$n^{1/3}\{F_n^{(1)}(z_0)-F_0(z_0)\}f_0(z_0)^{-1/3}\left\{2\sum_{i=0}^m\frac{(g(a_i)-g(a_i-))^2}{h(z_0+a_i)}\right\}^{1/3}\stackrel{\cal D}{\longrightarrow}2Z,$$

where $Z$ is the last time that standard two-sided Brownian motion minus the parabola $y(t)=t^2$ reaches its maximum. This is the conjectured local limit behavior of the MLE for decreasing mixture densities.