Extension of the klt quasi-torsor theorem to reductive groups

Establish whether the theorem asserting that a pair (X,φ^{-1}Δ) is klt for a torus quasi-torsor φ:X→Y extends from algebraic tori to quasi-torsors under more general reductive algebraic groups G.

Background

The paper proves that if G is a torus and (Y,Δ) is a klt-pair, then the induced pair (X,φ^{-1}Δ) is also klt under the stated quasi-torsor hypotheses. The authors then identify the extension to general reductive groups as unresolved. Their motivic-integration approach reduces the issue to understanding the sets R_y of integral points in bounded regions of the Bruhat–Tits building of G as the point y varies, a behavior they describe as difficult to analyze.