  In this paper, we investigate the validity of synthetic curvature-dimension
bounds in the sub-Finsler Heisenberg group, equipped with a positive smooth
measure. Firstly, we study the measure contraction property, in short
$\mathsf{MCP}$, proving that its validity depends on the norm generating the
sub-Finsler structure. Indeed, we show that, if it is neither $C^1$ nor
strongly convex, the associated Heisenberg group does not satisfy
$\mathsf{MCP}(K,N)$ for any pair of parameters $K \in \mathbb{R}$ and $N \in
(1,\infty)$. On the contrary, we prove that the sub-Finsler Heisenberg group,
equipped with a $C^{1,1}$ and strongly convex norm, and with the Lebesgue
measure, satisfies $\mathsf{MCP}(0,N)$ for some $N \in (1,\infty)$.
Additionally, we provide a lower bound on the optimal dimensional parameter,
and we also study the case of $C^1$ and strongly convex norms. Secondly, we
address the validity of the curvature-dimension condition pioneered by Sturm
and Lott-Villani, in short $\mathsf{CD}(K,N)$. We show that the sub-Finsler
Heisenberg group, equipped with a $C^1$ and strongly convex norm, and with a
positive smooth measure, does not satisfy the $\mathsf{MCP}(K,N)$ condition for
any pair of parameters $K \in \mathbb{R}$ and $N \in (1,\infty)$. Combining
this result with our findings regarding the measure contraction property, we
conclude the failure of the $\mathsf{CD}$ condition in the Heisenberg group for
every sub-Finsler structure.
