Weak definable amenability conjecture for discrete definable groups

Let $G$ be a group definable in an arbitrary structure $M$. Write ${G^*}^{00}_M$ and ${G^*}^{000}_M$ for the model-theoretic connected components of $G^*$.

Weak definable amenability conjecture. If $G$ is weakly definably amenable, then

$${G^*}^{00}_M={G^*}^{000}_M.$$

This is the discrete specialization of the generalized definable topological amenability conjecture. The supplied text presents it while discussing that specialization and gives no resolution status for it.