Definable topological amenability conjecture with a definable neighborhood basis

Let $G$ be a topological group definable in a structure $M$, and suppose that the members of a basis of open neighborhoods of the identity element $e$ are definable.

Definable topological amenability conjecture. If $G$ is definably topologically amenable, then

$${G^*}^{00}_{\operatorname{def},\operatorname{top}}={G^*}^{000}_{\operatorname{def},\operatorname{top}}.$$

This is stated as a restriction of the generalized conjecture and still extends the preceding definable and topological conjectures. The supplied text does not state a resolution of this restricted claim.