A Llarull type theorem on complete non-compact manifolds

Let $3\le n\le 7$, $2\le k\le n-1$, and $m=n-k-1$. We prove that a complete, connected, noncompact spin manifold $(M^n,g)$ with scalar curvature $R_M\ge k(k-1)$ is isometric to ${\mathbb{S}}^k\times {\mathbb{T}}_{\mathrm{\Lambda }}^m\times \mathbb{R}$ if it admits a smooth proper map of nonzero degree to ${\mathbb{S}}^k\times {\mathbb{T}}^m\times \mathbb{R}$ whose spherical component is 1-Lipschitz. The flat torus in the conclusion is not necessarily isometric to the target torus.