The Dying Gasp was added in patch 5.0 of the Shadowbringers expansion.
The Dying Gasp becomes available at level 80 as part of the “Shadowbringers” quest, which is given by Alisaie – The Tempest (X:16, Y:30).
The trial has a minimum ilevel of 410 to enter and rewards you with:
Allagan Tomestone of Poetics: 10