  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 CFT's 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.
