I think Ludo's wand will stay as it is permanently since its current state is naturally linked to what Star's wand looked like when it broke. It formed its own thing now based around the original missing piece, after all.
For the same reason I think Star's wand is also permanently stuck in its form, because even though it slightly changed due to the rebuilding process, it's still missing a piece.
This theory hasn't been definitively confirmed or disproved, but Moon was able to hold the wand in Storm the Castle with nothing happening; same with Willoughby in Fetch. There could be other explanations offered for these (Moon was wearing gloves, dogs/mouths don't count as "holding," it only changes if you hold it with the intent to use it-- though Willoughby did try to use it multiple times so probably not-- etc.) but to me the "permanently frozen" explanation seems simplest.
Also in Quest Buy the nearly dead and nonfunctional wand didn't change when Ludo grabbed it, yet did for Lobster Claws when at full power.