Motivic or homological stability conjecture for Manin components

Let $\pi:\mathcal{X}\to B$ be a Fano fibration and let $M$ range over the Manin components of $\operatorname{Sec}(\mathcal{X}/B)$, with degree tending to infinity. Stability conjecture for Manin components. The Manin components exhibit motivic or homological stability as the degree increases.

The source describes this as a vague principle: possible formulations include convergence of renormalized classes in a localization of the Grothendieck ring or stabilization of étale cohomology. The correct formulation is explicitly said to remain a topic of ongoing research.