Positive scalar-curvature rigidity conjecture for small-curvature noncompact manifolds

Let $C_s,v_0>0$ and $n\geq 3$. Let $(M^n,g_0)$ be a complete non-compact Riemannian manifold. Assume that

$$C_{sob}(M)\geq C_s^{-1},$$ $$\operatorname{Vol}_{g_0}\left(B_{g_0}(x,r)\right)\leq v_0r^n\quad\text{for all }(x,r)\in M\times\mathbb{R}_+,$$ $$\left\|\operatorname{Rm}(g_0)\right\|_{L^{n/2}(M)}\leq \delta_0(n,C_s,v_0),$$

and $\mathcal{R}(g_0)\geq 0$. Positive scalar-curvature rigidity conjecture. There exists $\delta_0(n,C_s,v_0)>0$ such that every such $(M,g_0)$ is isometric to $\mathbb{R}^n$. The conjecture would give a rigidity theorem under Sobolev control, at-most-Euclidean volume growth, small $L^{n/2}$ curvature, and nonnegative scalar curvature, extending the positive-mass rigidity perspective discussed in the paper.