Well, I don't think that's true. Eventually, the plate will be higher density than the material available at the point of its construction, so it will need to

Well, I don't think that's true. Eventually, the plate will be higher density than the material available at the point of its construction, so it will need to take material from outside the plate's current area to fill it. As the plate extends further into lower-density regions, it will have to extend its materials gathering exercises further and further away from the actual construction site, so they'll take longer and longer to get there. Even if the plate is very low density indeed, eventually you'll run out of planet and have to start dismantling the moon - which is quite a long way away - unless the plate has a lower density than the volume of space the moon encloses.