  We define the Heisenberg Kakeya maximal functions $M_{\delta}f$,
$0<\delta<1$, by averaging over $\delta$-neighborhoods of horizontal unit line
segments in the Heisenberg group $\mathbb{H}^1$ equipped with the Kor\'{a}nyi
distance $d_{\mathbb{H}}$. We show that $$ \|M_{\delta}f\|_{L^3(S^1)}\leq
C(\varepsilon)\delta^{-1/3-\varepsilon}\|f\|_{L^3(\mathbb{H}^1)},\quad f\in
L^3(\mathbb{H}^1),$$ for all $\varepsilon>0$. The proof is based on a recent
variant, due to Pramanik, Yang, and Zahl, of Wolff's circular maximal function
theorem for a class of planar curves related to Sogge's cinematic curvature
condition. As an application of our Kakeya maximal inequality, we recover the
sharp lower bound for the Hausdorff dimension of Heisenberg Kakeya sets of
horizontal unit line segments in $(\mathbb{H}^1,d_{\mathbb{H}})$, first proven
by Liu.
