Establish conservativity of the motivic nearby cycles functor

Establish the conservativity of the motivic nearby cycles functor used in the comparison of equal- and mixed-characteristic Hecke stacks.

Background

Motivic nearby cycles are a central ingredient in the paper's proof: they transport universally locally acyclic motives from the generic fiber of the Bando family to its special fibers. The argument is deliberately designed not to require conservativity of this functor.

The authors nevertheless refer to conservativity as conjectural, so proving it would resolve an explicitly identified unresolved issue in the motivic formalism and could simplify or strengthen comparison arguments of this type.

References

The ramified case requires uniform control of Pappas--Zhu group schemes throughout Bando's family, whereas the extension to motivic sheaves uses the motivic nearby cycles functor without ever relying on the conjectural conservativity of that functor.

— Parahoric motivic Hecke categories in equal and mixed characteristic  (2609.05123 - Bartling et al., 4 Sep 2026) in Section 1, paragraph beginning 'The purpose of this paper is to establish such a comparison'