[D] How is the MDP Homomorphism approach in the below paper different from RL with an Embedding Space like for example CURL?

I'm reading: Plannable Approximations to MDP Homomorphisms: Equivariance under Actions (arxiv: https://arxiv.org/pdf/2002.11963.pdf) and I'm not understanding how this approach differs from RL training in embedding space, for example, CURL. I do understand that they seem to be training both in the base space and the embedding space, but I'm not sure what this buys them. Thanks for your help!

embedding machinelearning rl space