Contractible definable neighborhood basis for the infinitesimal subgroup

Let $G$ be a definably compact group, let $M_0$ be the parameter structure over which $G$ is defined, and let $G^{00}$ denote the infinitesimal subgroup. Contractibility conjecture. There is a decreasing sequence of $M_0$-definable, definably contractible open subsets $X_i\subseteq G$ such that

$$G^{00}=\bigcap_{i\in\operatorname{\mathbb N}}X_i,$$

with

$$X_0\supset X_1\supset X_2\supset\cdots.$$

Such a sequence would provide a contractible definable neighborhood basis for $G^{00}$ and is used in the source's discussion of contractibility and the resulting cohomological consequences.