Existence of a dfg subgroup with definable, definably compact quotient over C((t))

Ascertain, up to finite index (in the virtual sense), whether for every group G definable in C((t)) there exists a definable subgroup H ≤ G with a definably f-generic (dfg) type such that the quotient G/H is definable in the language of valued rings (not merely interpretable) and is definably compact.

Background

The conjecture aims to ensure the existence of a well-behaved definable quotient by a dfg subgroup, strengthening interpretability to definability and imposing definable compactness on the quotient.

This would provide a robust structural decomposition of definable groups over C((t)) into a dfg part and a definably compact quotient, aligning with goals in the model theory of valued fields and facilitating further analysis.