  We initiate the study of synthetic curvature-dimension bounds in sub-Finsler
geometry. More specifically, we investigate the measure contraction property
$\mathsf{MCP}(K, N)$, and the geodesic dimension on the Heisenberg group
equipped with an $\ell^p$-sub-Finsler norm. We show that for $p\in(2,\infty]$,
the $\ell^p$-Heisenberg group fails to satisfy any of the measure contraction
properties. On the other hand, if $p\in(1,2)$, then it satisfies the measure
contraction property $\mathsf{MCP}(K, N)$ if and only if $K \leq 0$ and $N \geq
N_p$, where the curvature exponent $N_p$ is strictly greater than $2q+1$ ($q$
being the H\"older conjugate of $p$). We also prove that the geodesic dimension
of the $\ell^p$-Heisenberg group is $\min(2q+2,5)$ for $p\in[1,\infty)$. As a
consequence, we provide the first example of a metric measure space where there
is a gap between the curvature exponent and the geodesic dimension.
