Definable compact quotient by a definable dfg subgroup over $\mathbb{C}((t))$

Let $G$ be a group definable in $\mathbb{C}((t))$. Interpret all assertions in the virtual sense. A subgroup is dfg when it has the dfg property used by the source.

Definable compact quotient conjecture. There is a $\mathbb{C}((t))$-definable dfg subgroup $H$ of $G$ such that $G/H$ is definable, rather than merely interpretable, and definably compact.

The source gives no resolution evidence for this conjecture; it is intended to provide a definable compact quotient after passing to a suitable virtual subgroup.