A demo for The Legend of Heroes: Trails of Cold Steel III has appeared on PlayStation Network. It seems to be live only on European stores [UK] at the moment, but an NA release should be within the next day or so. [Update: Now up on NA store.]
NIS America had previously announced plans to release a demo, although it was not dated at the time.
The demo seems to contain the game's prologue, and progress will be transferred to the full game. The store description does make a note about an early game trophy, which you can find below.
Rean Schwarzer uncovers a dark plot that threatens his homeland. To face their enemies, he must prepare a new generation of heroes as an instructor at a new branch campus and guide them towards victory.
NOTE: No trophies will be acquired in this demo version. In order to ensure you are able to acquire the trophy for completing the prologue in the full game, please follow these steps:
Create a save in the demo right before its end boss.
Transfer that save to the full game.
Defeat the boss in the full game to acquire the trophy.
The Legend of Heroes: Trails of Cold Steel III is set to release on October 22 in North America & Europe and October 29 in Oceania for PlayStation 4.