Universality of persistent homology distributions for uniform measures

Let $X\subseteq\mathbb{R}^m$ be compact with positive Lebesgue measure, and let $\mu$ be Lebesgue measure restricted to $X$ and rescaled to have total mass one. Let $\hat{F}^{(i)}_n$ denote the empirical cumulative distribution function of the lengths of the $i$-dimensional persistent homology intervals of an i.i.d. sample of $n$ points. Uniform persistent homology distribution conjecture. The limit

$$\hat{F}^{(i)}(t)=\lim_{n\to\infty}\hat{F}^{(i)}_n(n^{-1/m}t)$$

exists and depends only on $m$, $i$, and the volume of $X$. The paper states that this claim is false for non-uniform measures; its validity in the uniform case is not established generally.