  We show that the communication cost of quantum broadcast channel simulation
under free entanglement assistance between the sender and the receivers is
asymptotically characterized by an efficiently computable single-letter formula
in terms of the channel's multipartite mutual information. Our core
contribution is a new one-shot achievability result for multipartite quantum
state splitting via multipartite convex splitting. As part of this, we face a
general instance of the quantum joint typicality problem with arbitrarily
overlapping marginals. The crucial technical ingredient to sidestep this
difficulty is a conceptually novel multipartite mean-zero decomposition lemma,
together with employing recently introduced complex interpolation techniques
for sandwiched R\'enyi divergences.
  Moreover, we establish an exponential convergence of the simulation error
when the communication costs are within the interior of the capacity region. As
the costs approach the boundary of the capacity region moderately quickly, we
show that the error still vanishes asymptotically.
