Well, Asylon was a colony of Caerwyn, and according to the treaty should inherit its claims. 
Depends on your interpretation of the legalese.
"In the event of a secession from one of the signing realms, the new realm (or Child Realm) is considered to have the same Legal Claims as their Parent Realm (the realm they seceded from), until such a time as this treaty is rewritten."
Technically, Asylon was formed before this treaty. Since then, Caerwyn
has had a secession... two, in fact. Via to Asylon and Itaulond. Presumably the duchy of Via would have claim to its own regions (which accompanied it) and the duchy of Itau would be the legal successor to its own regions (which accompanied it). Caerwyn could turn over legal claims to Asylon,
if Caerwyn still existed. But, in its demise, to my knowledge, it never formally transfered legal claims.
Thus, while this treaty does not establish a clear definition of what to do with claims in the event of a realm's destruction, it does establish a few principles which I shall refer to as "Nonexpansion," "Entity, and "Heritability."
Nonexpansion- Even in the event of secessions and region revolts, claims do not change. The treaty is evidently framed to prevent "claim creep" wherein one realm gradually secures more and more claims. It is an inflexible claim regime.
Heritability- Claims are passed down to succeeding constituent entities. Caerwyn's claims would be assumed by successor states, meaning the Duchy of Via but, probably more properly, Itaulond. Itaulond probably has the strongest claim to Caerwyn's share of the treaty.
Entity- Claims can evidently be traded and exchanged, meaning they are not to be regarded as inalienable.
Under nonexpansion, Asylon would clearly have an issue: they are clearly engaging in "claim creep," presuming to claim even lands directly under Terran's influence which they can never hope to control. Under heritability, Asylon never formally become Caerwyn's heir. But, the best they can reasonably hope for is that they can claim the Duchy of Via legally, while Itaulond gets the Duchy of Itau, and other regions become "free lands." This is the strongest argument for Asylon's claims. A different interpretation of the treaty would suggest that Itau maintains claims to all lands Caerwyn ever claimed, as it is the only "Child Realm," and Via's allegiance change could threaten Nonexpansion. Under Entity, we could regard claims as being "fiat assets," whereby the destruction of their holding entity (namely, Caerwyn) would destroy the asset as well, so that all of Caerwyn's lands, even in the Duchy of Via and Itaulond, would technically be "free lands" under the treaty, though obviously not under any practical assessment.
