One-half reflected entropy is not a lower bound for entanglement of purification

In recent work, Akers et al. proved that the entanglement of purification E_p(A:B) is bounded below by half of the q-Rényi reflected entropy S_R^(q)(A:B) for all q≥2, showing that E_p(A:B) = 1/2 S_R^(q)(A:B) for a class of random tensor network states. Naturally, the authors raise the question of whether a similar bound holds at q = 1. Our work answers that question in the negative by finding explicit counter-examples, which we arrive at through numerical optimization. Nevertheless, this result does not preclude the possibility that restricted sets of states, such as CFT states with semi-classical gravity duals, could obey the bound in question.