Injective Simplicial Maps of the Complexes of Curves of Nonorientable Surfaces

Let $N$ be a compact, connected, nonorientable surface of genus $g$ with $n$ boundary components, and $\mathcal{C}(N)$ be the complex of curves of $N$. Suppose that $g + n \leq 3$ or $g + n \geq 5$. If $\lambda : \mathcal{C}(N) \rightarrow \mathcal{C}(N)$ is an injective simplicial map, then $\lambda$ is induced by a homeomorphism of $N$.