It's quite easy to switch LLM api, so you can just transition to a competitor. Competition between AI providers is quite fierce, I don't see them setting up a cartel anytime soon. And open source models are not that far beyond commercial ones.
But how I see it is that for solving KC in full generality you'll have to:
- Start with the program that explicitly returns the original string. Let's say it has length N
- run all possible programs that are shorter than N (just try all combinations of characters)
- look at the results and pick the shortest program that compiles and outputs the original string
The problem there is that you have to wait for all programs to end, and you don't know if they will end or not. So you have a problem that's equivalent to the halting problem (and that's not solvable) (and the halting problem is loosely related to the interesting number problem).
(This is not a proof and I don't have a background in the field btw)