Exhaustion of $\mathcal{C}(N)$ via rigid expansions

Let $N$ be a connected closed non-orientable surface of genus at least 6. In this work we prove that there exists a finite subgraph $\mathfrak{X}$ (Irmak's finite rigid set from ``Elmas Irmak. Exhausting curve complexes by finite rigid sets on nonorientable surfaces. J.Topol.Anal., 16(2):261--289, 2024'') such that any graph endomorphism $\varphi$ of $\mathcal{C}(N)$ whose restriction to $\mathfrak{X}$ is (locally) injective, $\varphi$ is induced by a homeomorphism of $N$. To prove this, we first prove that $\mathfrak{X}$ and its rigid expansions exhaust $\mathcal{C}(N)$.