The asymptotic arithmetic fibration conjecture for L-targets

Let $V$ be a weakly ${\cal L}$-saturated variety. Assume that $V$ does not contain an open Zariski dense strongly ${\cal L}$-saturated subset $U\subset V$. The set of ${\cal L}$-targets of $V$ is the collection used in the paper's counting strategy.

Asymptotic arithmetic fibration conjecture. The set of ${\cal L}$-targets of $V$ forms an asymptotic arithmetic ${\cal L}$-fibration.

This conjecture is the second reduction step in the paper's proposed strategy for understanding the asymptotics of $N(V,{\cal L},B)$. The supplied text gives no resolution status.