The authors consider phylogeny estimation under a two-state model of sequence evolution by site substitution on a tree. They demonstrate the assertion that for any fixed $k$, no statistically consistent phylogeny estimation is possible from the $k$-mer counts of the entire leaf sequences alone. This is accomplished as follows. \par The authors consider a symmetric substitution model on phylogenies, also known as the Cavender-Ferris-Neyman (CFN) model, for binary sequences of fixed length $m$. The CFN model on a single edge of a metric tree is a continuous-time Markov process with state space $\{0,1\}^m$ such that the $m$ coordinates switch states independently according to a continuous-time two-state Markov process which runs at unit rate and has the same jump rates for each coordinate. \par The authors are interested in this process on a rooted metric tree $T$ for which each edge $e=(u,v)$ has an associated time $\ell_e$. The root vertex $\rho$ is assigned an initial state ${\cal X}_\rho\in\{0,1\}^m$, drawn from the uniform distribution on $\{0,1\}^m$. Moving away from the root, along each edge $e=(u,v)$, one runs the CFN process described above for a time $\ell_{(u,v)}$ with initial state $\Cal{X}_u$. Such processes along different edges starting at $u$ are conditionally independent given $\Cal{X}_u$. Denote by $\Cal{X}_t$ the resulting state at $t\in e$. The full process $\{\Cal{X}_t\}_{t\in T}$ is called the CFN model on tree $T$. In particular, this setup ultimately assigns a randomly generated $m$-long binary string to each leaf of $T$. \par For a positive integer $k$, a $k$-mer is a $k$-long binary string, i.e., $y\in\{0,1\}^k$. Let $m\ge k$ and for an arbitrary $m$-long binary string $\sigma=\{\sigma_i\}_{i=1}^m$, let $f_\sigma(y)$ denote the number of times that the $k$-mer $y$ appears in $\sigma$ as a consecutive substring. That is, $$ f_\sigma(y)=\sum_{i=0}^{m-k}{\bf 1}\{(\sigma_{i+1},\dots,\sigma_{i+k})=y\}. $$ The frequency vector of $k$-mers in $\sigma$ is the vector $f_\sigma=(f_\sigma(y))_{y\in\{0,1\}^k}$, where the numerical position of the coordinate corresponding to a given $y$ is 1 plus the integer whose binary representation corresponds to $y$. \par Let $T_1$ and $T_2$ be two trees, each with $n\ge 3$ leaves, and let $ {\cal L}_m^{(i)}$ denote the law of the $k$-mer frequencies of the ($m$-long) leaf sequences of $T_i$ which are generated by the above Markov process of node sequences. The authors prove that there exist distinct trees, $T_1\ne T_2$, such that the total variation distance between the leaf frequency laws satisfies $\lim_{m\to\infty} \| {\cal L}_m^{(1)}-{\cal L}_m^{(2)}\|_{TV}<1$. In addition, the trees $T_i$ can be chosen to be independent of $k$. In other words, there is no statistical test based on $k$-mer counts alone which can distinguish between $T_1$ and $T_2$ with success probability tending to unity as the sequence length $m\to\infty$. The authors stress that their result does not apply to block decomposition methods. \par The paper includes an extensive literature review, in essence the above account and a sketch of the proof, the twelve-page proof, remarks and an appendix about total variation results used in the proof.