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In recent work, Akers et al. proved that the entanglement of purification$E_p(A:B)$ is bounded below by half of the $q$-R\'enyi reflected entropy$S_R^{(q)}(A:B)$ for all $q\geq2$, showing that $E_p(A:B) = \frac{1}{2}S_R^{(q)}(A:B)$ for a class of random tensor network states. Naturally, theauthors raise the question of whether a similar bound holds at $q = 1$. Ourwork answers that question in the negative by finding explicitcounter-examples, which we arrive at through numerical optimization.  Nevertheless, this result does not preclude the possibility that restrictedsets of states, such as CFT states with semi-classical gravity duals, couldobey the bound in question.

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