RHYTHM OF WAR SPOILERS!
TL;DR: Adolin and Shallan release Ba-Ado-Mishram and revive all the Deadeyes.
Some Tinfoil And Some Facts
After completing WoR, I had a big question I never asked 'cause I thought Stormfather was exaggerating what had happened... but I'll come back to this point in a min.
What are Deadeyes? Spren whose bonded radiant broke their oaths. But isn't the cost too high? Humans who can't keep their oath lose their Surgebinding at the cost of a dead spren?
After reading RoW, we know what happens to a spren whose Radiant broke their oaths after the Recreance. Testament died after Recreance when Shallan broke her oaths... err, truths?
Now back to my first point. Stormfather told Kaladin that he killed Syl. Back then I assumed he was lying and Syl will just return to Shadesmar after their bond breaks completely, which never happened. Now that I think about it, Stormfather wasn't lying. Syl would have died if her bond with Kal broke completely.
Read This Before Conclusion
Syl said this about Honorblades:
If a Spren reqiure checks, will it just die if the the requirements are not met. I don't think so.
Maya agrees with me:
Spren thought they will suffer from pain, not death.
And Kalak says:
I certainly think this is related to formation of Deadeyes after a Radiant break oaths after the Recreance
The Conclusion
Before Ba-Ado-Misharm was imprisoned, a spren would not die if their Radiant broke their oaths. They would just be sent back to Shadesmar.
And ever since the Recreance, any broken bond results to death of the spren.
But why Ba-Ado-Mishram is so important? How can an Unmade scar Roshar in such ways? I think Ba-Ado, and all the unmade for the fact, were once of Honor and Cultivation with important roles in Rosharan Spiritual ecology. Later currupted by Oduium. But that's another theory for another day.
Adolin and Shallan's arc is now going towards reviving the Deadeyes. Maya and Adoin have been theorized for long. Now we have Shallan and Testament too.
I think Adolin and Shallan's next quest will be to find Ba-Ado and free her. In freeing her, they will revive the Deadeyes.
Another Tinfoil Titbit
Ba-Ado-Mishram was once tasked to check Surgebinding in ways that it remains safe and whatever happened to Ashyn is not repeated.
She kept doing it even after Odium Unmade her 'cause Odium don't want to destroy Roshar either.
After her imprisonment, Radiants and their spren realized that Nahel bonds are no longer safe and decided to break them.