For any state of a quantum field theory, the entanglement entropy (EE) of subsystems satisfies two inequalities known as subadditivity (SA) and strong subadditivity (SSA). In holography, states with a smooth classical geometric dual are constrained further. For static states, the EE is given by the Ryu-Takayanagi (RT) formula, which has been shown to obey both SA and SSA, as well as an infinite family of higher inequalities. We address the question of whether holographic entropy inequalities obeyed in static states (by the RT formula) are obeyed in time-dependent states (by the HRT formula). Working in the context of 2+1 dimensional AdS3 quotient spacetimes, we numerically test all known inequalities for the HRT formula founding no counterexamples. We also introduce a new formulation of the HRT formula (known as minimax), and use it to explore the above question in full generality. The minimax formulation will imply a new decomposition of the bulk spacetime which may be helpful in covariantizing both the holographic entropy cone and tensor network models of holography.