Classification of dfg groups over C((t)) as virtually split solvable algebraic groups

Determine whether every group G definable in C((t)) that admits a definably f-generic (dfg) type is virtually a C((t))-split solvable algebraic group; that is, ascertain whether G contains a finite-index subgroup that is a solvable algebraic group split over C((t)).

Background

Definably f-generic (dfg) groups play a key role in the modern theory of definable groups, with parallels drawn from p-adically closed fields where dfg often correlates with solvable algebraic structure.

The conjecture seeks to extend such structural classifications to groups definable over C((t)), positing that dfg implies being virtually a split solvable algebraic group over the base field.