  The RT formula for static spacetimes arising in the AdS/CFT correspondence
satisfies inequalities that are not yet proven in the case of the HRT formula,
which applies to general dynamical spacetimes. Wall's maximin construction is
the only known technique for extending inequalities of holographic entanglement
entropy from the static to dynamical case. We show that this method currently
has no further utility when dealing with inequalities for five or fewer
regions. Despite this negative result, we propose the validity of one new
inequality for covariant holographic entanglement entropy for five regions.
This inequality, while not maximin provable, is much weaker than many of the
inequalities satisfied by the RT formula and should therefore be easier to
prove. If it is valid, then there is strong evidence that holographic
entanglement entropy plays a role in general spacetimes including those that
arise in cosmology. Our new inequality is obtained by the assumption that the
HRT formula satisfies every known balanced inequality obeyed by the Shannon
entropies of classical probability distributions. This is a property that the
RT formula has been shown to possess and which has been previously conjectured
to hold for quantum mechanics in general.
