Nope! In LD, Peridot and Jasper came to start the kindergarten back up, along with eradicate the Crystal Gem menace. While they were at it, the authority wanted them to produce gemlings because they had nothing else to do aboard the ship on the way to and from earth. However, when the report came back that the ship had crashed, the authority assumed them smashed with the rubble and Earth a lost cause. They weren’t going to waste any more resources on a couple traitors when they had to focus on other things.
I wish I could put this in the story itself, but since it focuses on the CGs, Peri, and their offspring, I don’t know how to put it in. :/
But, I’m glad you asked! Questions will always been answered, unless they’re spoiler-y.