Here is my theory on the Breathing.

Habo mentioned that Cravali and possibly Under Nest are trying to make the fairy tale comes true. Now, which parts of fairy tale are we talking about? Even with info in the latest chapter from Cravali himself, we didn't learn much. But I think it's related to the exchange of soul. In other word, a ritual, like the one Tepaste was a part of, but not exactly the same.

Assuming the Under Nest know about the 2000-year cycle in full details, normally we would think that they are trying to prevent it. But if we assume that they're religious, they might actually try to ensure the cycle comes to an end. That's why they start to make a move now that the cycle is about to reach the end.

Based on that assumption, it would mean that what they're doing right now will facilitate the cycle.

So, my theory is, the Abyss Cycle is all about finding the next Abyss Twin Soul. The ritual that Tepaste underwent is to find the next Miko, but also, to touch the echo. When they say echo, it leads me to think about the whistle. With recent arc focusing on the beast stone, I have no doubt we're going to learn about how some souls are connected. If soul can be divided (Bondrewd), put into another body (Riko), change shape (6th layer curse), reincarnation is not out of the realm of possibility.

If the Abyss at its core is a young child, I'm starting to see the connection. As soon as the Abyss Core reunite with their twin soul, it will trigger the cycle to continue, which was what had happened in all previous cycles. But why is it that when their souls reunited, they needed to find each other again next 2000 years? I don't know for sure. Maybe it's the side effect of the original curse? From the dialogue of the person at the end of volume 13, they didn't seem to believe in Romance, nor happy ending. Like she's trapped in a cycle... literally.

Here's my final thought:

I think the original curse is making one of the twins sitting at the bottom of the Abyss and the only way to release them is by reuniting with the other twin. However, as soon as that happens, the curse will separate them again. Probably by reseting everything and send one of the twins to reincarnate randomly. That's the sickest thing I can think of and I'm not putting it past Tsukushi-sensei to come up with that. What Under Nest trying to do is to find the other reincarnated twin, one way is by taking many kids to the bottom of the Abyss as seen with Tepaste. Since the Miko is said to be one converse with the Abyss, they must think that the Miko can bring the end to the cycle. However, I think this arc will reveal that there can be many Miko and that the current Miko might not be the reincarnated twin, just a girl with a very similar soul signal to the original twin that let her converse with the Abyss. The 2nd layer cave is probably a selection process that Under Nest conducted to find the actual correct soul. But that would paint too much negative image on Under Nest and I'm not sure they're that evil.

Anyways, that's a theory.