Conjectured Chernoff limit for the uniform deconvolution MLE

Let $F_0$ have a continuous positive density $f_0$ on $(0,M)$, where $M>0$, and let $t_0\to(a,b)$ be an interior point. Let $\hat F_n$ be the nonparametric maximum likelihood estimator of $F_0$. Let $W$ be two-sided Brownian motion on $\mathbb R$ originating from zero. Conjectured Chernoff limit. The normalized estimation error is conjectured to satisfy

$$\frac{n^{1/3}\{\hat F_n(t_0)-F_0(t_0)\}}{\left(4f_0(t_0)F_0(t_0)(1-F_0(t_0))\right)^{1/3}} \stackrel{d}{\longrightarrow} \operatorname*{argmin}_{t\in\mathbb R}\left\{W(t)+t^2\right\}.$$

This conjectures a cube-root asymptotic distribution of Chernoff type for the nonparametric MLE in the uniform deconvolution model. The surrounding discussion contrasts the fixed and mixed models and indicates that the asserted behavior remains a conjecture in the uniform case.