Every lossless “compression” algorithm makes some strings shorter and other strings longer, such that the average space savings against all possible strings is neutral at best. This is a fundamental consequence of the way information works — there’s no way to losslessly map (say) 8 bits of input into 7 bits of output.

The goal of compression, then, is to make common strings shorter while making uncommon strings longer. You can think of, say, UTF-8 as a simple compression algorithm: it would take 20 bits per character to encode all Unicode code points with a fixed-width encoding, but UTF-8 takes advantage of the fact that the most commonly used characters have a lot of leading zeroes (at least in English). So the characters of the English alphabet require only 8 bits to encode, but uncommon characters require up to 32 bits.

Thus, I would expect an LLM-based compression algorithm to do well on strings that were common in its training data, and make strings that were uncommon or absent slightly longer. If it did not do that, it would not be a lossless compression algorithm.