Limit-measure uniqueness conjecture for decorated étale group schemes

Let $\mathcal{E}$ be the space of decorated étale group schemes considered in the paper, and let $mu_g$ denote the probability measure associated with genus or size parameter $g$. For an element of $\mathcal{E}$, consider the number of surjections from a random decorated group scheme to it.

Limit-measure conjecture. The measures $mu_g$ converge to a measure $mu$ on $\mathcal{E}$ such that the expected number of surjections from a $mu$-random element to every element of $\mathcal{E}$ is $1$; moreover, this property uniquely characterizes $mu$.

This conjecture asserts that the moments determine the limiting distribution. The paper presents it as an analogue of corresponding uniqueness results for Cohen–Lenstra measures and proves related results in special cases, but leaves the full decorated setting open.