Equality of connected components and bounded-index components over $\mathbb{C}((t))$

Let $G$ be a group definable in $\mathbb{C}((t))$. Write $G^0$ for the connected component and $G^{00}$ for the smallest type-definable subgroup of bounded index.

Connected-component conjecture. One has

$$G^0=G^{00}.$$

Consequently, $G$ is virtually an open subgroup of an algebraic group over $\mathbb{C}((t))$.

The source presents this as a conjecture and compares the consequence with Corollary 4.3 of the work cited as JY-2; no resolution is supplied.