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 $q\ge 2$, showing that $E_p(A:B)=\frac{1}{2}{S}_R^{(q)}(A:B)$ 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.