The connected-component inclusion conjecture for group extensions

Let $(\widetilde{G^*},A^*,G^*)$ be in Situation $(00)$, and let $B$ be the parameter set used to define the connected components. Suppose

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

Connected-component inclusion conjecture. Then

$$\widetilde{G^*}^{00}_B\cap A^*\subseteq A^*_1\iff \widetilde{G^*}^{000}_B\cap A^*\subseteq A^*_1.$$

This is presented as an equivalent reformulation of the subsequent conjecture and is stated without a resolution; the paper notes that related implications are known under additional hypotheses, including countability and $A^*_1\subseteq\widetilde{G^*}^{000}_B\cap A^*$, while the unrestricted form remains open.