We argue that holographic CFT states require a large amount of tripartite entanglement, in contrast to the conjecture that their entanglement is mostly bipartite. Our evidence is that this mostly-bipartite conjecture is in sharp conflict with two well-supported conjectures about the entanglement wedge cross section surface $E_W$. If $E_W$ is related to either the CFTs reflected entropy or its entanglement of purification, then those quantities can differ from the mutual information at $mathcal{O}(frac{1}{G_N})$. We prove that this implies holographic CFT states must have $mathcal{O}(frac{1}{G_N})$ amounts of tripartite entanglement. This proof involves a new Fannes-type inequality for the reflected entropy, which itself has many interesting applications.
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