Intrinsic characterization conjecture for codimension-one embeddings of nonnegatively curved spheres

Let $g$ be a smooth metric of non-negative sectional curvature and positive scalar curvature on $\mathbb S^n$ which is locally isometrically embeddable in $\mathbb R^{n+1}$.

Intrinsic characterization conjecture. $(\mathbb S^n,g)$ admits a smooth global isometric embedding

$$\mathbf X\colon(\mathbb S^n,g)\to\mathbb R^{n+1}.$$

This conjecture asks for an intrinsic characterization of the non-negatively curved metrics on spheres that admit codimension-one isometric embeddings. The local embeddability assumption is necessary, and the authors emphasize that it is a non-trivial restriction in dimensions $n\geqslant 3$; no resolution is supplied here.