Before revision:  In particular, we show that when the RKHS is infinite dimensional , any lower bound must depend on the cardinalities of the persistence diagrams, and that when the RKHS is finite dimensional, finding a bi-Lipschitz embedding is impossible, even when restricting the persistence diagrams to have bounded cardinalities.
After revision:  In particular, we show that for several stable embeddings into infinite-dimensional Hilbert spaces defined in the literature , any lower bound must depend on the cardinalities of the persistence diagrams, and that when the RKHS is finite dimensional, finding a bi-Lipschitz embedding is impossible, even when restricting the persistence diagrams to have bounded cardinalities.