Reflected entropy is not a correlation measure

By explicit counterexample, we show that the "reflected entropy" defined by Dutta and Faulkner is not monotonically decreasing under partial trace, and so is not a measure of physical correlations. In fact, our counterexamples show that none of the R\'enyi reflected entropies $S_{R}^{(\alpha)}$ for $0 < \alpha < 2$ is a correlation measure; the usual reflected entropy is realized as the $\alpha=1$ member of this family. The counterexamples are given by quantum states that correspond to classical probability distributions, so reflected entropy fails to measure correlations even at the classical level.