  In this paper, without assuming that manifolds are spin, we prove that if a
compact orientable, and connected Riemannian manifold $(M^{n},g)$ with scalar
curvature $R_{g}\geq 6$ admits a non-zero degree and $1$-Lipschitz map to
$(\mathbb{S}^{3}\times
\mathbb{T}^{n-3},g_{\mathbb{S}^{3}}+g_{\mathbb{T}^{n-3}})$, for $4\leq n\leq
7$, then $(M^{n},g)$ is locally isometric to
$\mathbb{S}^{3}\times\mathbb{T}^{n-3}$. Similar results are established for
noncompact cases as $(\mathbb{S}^{3}\times
\mathbb{R}^{n-3},g_{\mathbb{S}^{3}}+g_{\mathbb{R}^{n-3}})$ being model spaces
(see Theorem \ref{noncompactrigidity1}, Theorem \ref{noncompactrigidity2},
Theorem \ref{noncompactrigidity3}, Theorem \ref{noncompactrigidity4}). We
observe that the results differ significantly when $n=4$ compared to $n\geq 5$.
Our results imply that the $\epsilon$-gap length extremality of the standard
$\mathbb{S}^3$ is stable under the Riemannian product with $\mathbb{R}^m$,
$1\leq m\leq 4$ (see $D_{3}$. Question in Gromov's paper \cite{Gromov2017},
p.153).
