The type-definable subgroup inclusion conjecture for group extensions

Let $(\widetilde{G^*},A^*,G^*)$ be in Situation $(0)$, and let $B$ be the parameter set used to define the connected components. Suppose

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

Let $H$ be a $B$-invariant subgroup of $\widetilde{G^*}$ of bounded index such that $H\cap A^*$ is type-definable. Type-definable subgroup inclusion conjecture. Then

$$\widetilde{G^*}^{00}_B\cap A^*\subseteq H\cap A^*.$$

This conjecture is stated as equivalent to the preceding conjecture and generalizes the nearby corollary; the supplied text gives no proof or disproof, so the assertion remains open.