A Faint Hope

A Faint Hope is Disc 4, Track 17 on the Xenoblade2 Original Soundtrack. Its melody is based off a motif from Elysium, in the Blue Sky. It plays in the following cutscenes in Xenoblade Chronicles 2:
 * Vandham tells Rex about the nature of war in the aftermath of Akhos's attack on Garfont.
 * Lila provides Poppi with her energy in order to fully awaken her ether furnace (flashback).
 * Pyra shows Eulogimenos Addam's final message to his people.
 * Azurda tells Rex that he will show the way to the 3rd Aegis Sword.
 * Rex speaks with Addam's memory in the dream of Elysium.
 * The Architect sends off the party after their long chat.

It also plays in the following cutscenes in Xenoblade Chronicles 2: Torna ~ The Golden Country:
 * Addam explains the original form of the Tornan titan.
 * The Tornan king requests the party's continued assistance in fighting Malos, and offers Lora knighthood.