Ok, I'm gonna post my theory. Cairngorm probably suspected that achmea was hiding important information from the gems in order to get them to do his bidding. Cuz you know, telling people half truths and lies let's them fill in the gaps on their own. But in the mid chapters, Phos relearned the story from the admirablis about the humans and talked to ghost about it, which transferred to Cairngorm. Cairngorm, being kind of a b**** on Earth suddenly turns into a bubbly love struck dolt as a way in with the leader. Bc, let's be honest, aside from Cairngorm tossing their arm in the blender, achmea came on pretty strong. I think this plan is to get achmea to reveal all the secrets to them and explain it to the gems on the moon. That way, if it turns out to be something crazy, the gems are the force behind enemy lines. Plus, they have learned their science and how to use their technology. As for Rutile, I feel like she's going to have like an epiphany from holding on to the gem fragment for so long or something. And considering that she is considered the foremost expert on inclusion filled gems, she might learn something that can help Phos condition.
Just a theory I've had for a bit.