  Let $K$ be a convex body in $\mathbb{R}^n$ (i.e., a compact convex set with
nonempty interior). Given a point $p$ in the interior of $K$, a hyperplane $h$
passing through $p$ is called barycentric if $p$ is the barycenter of $K \cap
h$. In 1961, Gr\"{u}nbaum raised the question whether, for every $K$, there
exists an interior point $p$ through which there are at least $n+1$ distinct
barycentric hyperplanes. Two years later, this was seemingly resolved
affirmatively by showing that this is the case if $p=p_0$ is the point of
maximal depth in $K$. However, while working on a related question, we noticed
that one of the auxiliary claims in the proof is incorrect. Here, we provide a
counterexample; this re-opens Gr\"unbaum's question. It follows from known
results that for $n \geq 2$, there are always at least three distinct
barycentric cuts through the point $p_0 \in K$ of maximal depth. Using tools
related to Morse theory we are able to improve this bound: four distinct
barycentric cuts through $p_0$ are guaranteed if $n \geq 3$.
