  For any dimension $n \geq 2$, we consider the maximal directional Hilbert
transform $\mathscr{H}_U$ on $\mathbb R^n$ associated with a direction set $U
\subseteq \mathbb S^{n-1}$: \[ \mathscr{H}_Uf(x) := \frac{1}{\pi} \sup_{v \in
U} \Bigl| \text{p.v.} \int f(x - tv) \, \frac{dt}{t}\Bigr|.\] The main result
in this article asserts that for any exponent $p \in (1, \infty)$, there exists
a positive constant $C_{p,n}$ such that for any finite direction set $U
\subseteq \mathbb S^{n-1}$, \[||\mathscr{H}_U||_{p \rightarrow p} \geq C_{p,n}
\sqrt{\log \#U}, \] where $\#U$ denotes the cardinality of $U$. As a
consequence, the maximal directional Hilbert transform associated with an
infinite set of directions cannot be bounded on $L^p(\mathbb{R}^{n})$ for any
$n\geq 2$ and any $p \in (1, \infty)$. This completes a result of Karagulyan,
who proved a similar statement for $n=2$ and $p=2$.
