Imagine that we have a tree with 1000 leaves and we want to have a Merkle proof that can prove the presence of, let's say 5 elements. The sparse Merkle multiproof will require an index for every non-leaf hash, so worst case around 30-40 indices (that's just an estimate, as every multiproof can have different sizes, depending on the location of the leaves). The compact Merkle multiproof on the other hand requires only 5 indices.
Unless I misunderstood something and we are not trying to minimize the size of whole proof, but just its structure - but I see no reason why it could be useful. But I can see there are two other papers: one on distributed Bloom filters [1] and the other on Bloom trees [2], maybe these can shed some light on it, I'll check later today.
[1] https://arxiv.org/pdf/1910.07782.pdf
[2] https://arxiv.org/pdf/2002.03057.pdf
EDIT: I haven't noticed you are one of the authors! I liked the paper, it is very clear and visualizes how Merkle (multi)proofs work in a nice way. The only thing that worries me is claim of significant reduce in the size without providing any hard data in that topic.
Current sparse Merkle multiproofs require an index and a hash inside the proof. So in our hypothetical example that would be 30-40 indices and 30-40 hashes. Only then can a recipient of the proof perform the operations in the right order to recreate the original merkle root. With our proposal on the other hand, for this hypothetical example, we would need 30-40 hashes, and only 5 indices for a recipient to be able to recreate the correct Merkle root.
Here's by the way a great repo that can create sparse Merkle multiproofs https://github.com/wealdtech/go-merkletree