Persistent homology scaling conjecture

Let $X_n$ be a random sample of $n$ points from a compact set $X\subseteq\mathbb{R}^m$, distributed according to a nonsingular probability measure $\mu$, where $m\geq 2$. Let $L^i(X_n)$ denote the sum of the lengths of the intervals in $i$-dimensional persistent homology. Persistent homology scaling conjecture. For all $0\leq i<m$, there is a constant $C\geq 0$, depending on $\mu$, $m$, and $i$, such that

$$L^i(X_n)=Cn^{(m-1)/m}$$

with probability one as $n\to\infty$. This extends the known $i=0$ scaling result implied by Steele's work; the higher-dimensional cases are presented as conjectural.