Strong universality of persistent homology ell-values

Let $\mathbb{S}$ be a sampling model in the class $\mathcal{U}$. For a complex type $\mathcal{T}$ and homological degree $k\geq 1$, let $\mathcal{L}_n$ be the empirical measure of the persistent $\ell$-values. Strong universality conjecture. For every $\mathbb{S}\in\mathcal{U}$, every $k\geq1$, and $\mathcal{T}\in\{\text{Čech},\text{Rips}\}$,

$$\lim_{n\to\infty}\mathcal{L}_n=\mathcal{L}^*,$$

where $\mathcal{L}^*$ is independent of $\mathbb{S}$, $\mathcal{T}$, and $k$. This conjecture formalizes the experimentally observed universality across iid, non-iid, dynamical, and real-data sampling models; the class $\mathcal{U}$ is intended to comprise a wide range of such models.