Construct Bando Grassmannians for wildly ramified groups

Construct an extension of a wildly ramified reductive group over a local field to a reductive group scheme over the Laurent polynomial ring O_F[u^{\pm1}] that can be used to define a Bando Grassmannian, thereby extending the parahoric motivic Hecke-category comparison beyond the tamely ramified case.

Background

The paper proves its equal- and mixed-characteristic comparison for connected reductive groups that are quasi-split and split over a tamely ramified extension. The construction relies on a Pappas–Zhu group scheme over O_F[u] and the associated Bando family of affine Grassmannians.

The authors identify wild ramification as a more serious unresolved obstruction: they do not know how to construct the required reductive group scheme over O_F[u^{\pm1}] for wildly ramified groups. Such a construction would be needed to define the corresponding Bando Grassmannian and to pursue an analogous comparison in the wildly ramified setting.