Marques–Neves width stability conjecture for rotationally symmetric three-spheres

Fix $D,V<\finite$. Suppose $(\mathbb S^3,g_k)$, $k=1,2,\ldots$, are Riemannian 3-spheres satisfying

$$\operatorname{Scal}_{g_k}\geqslant 6(1-k^{-1}),\qquad \operatorname{MinA}_{g_k}\geqslant 4\pi(1-k^{-1}),\qquad \operatorname{diam}_{g_k}(\mathbb S^3)\leqslant D,\qquad \operatorname{Vol}_{g_k}^3(\mathbb S^3)\leqslant V.$$

Marques–Neves' width stability conjecture. Then $(\mathbb S^3,g_k)$ converges in the Sormani–Wenger intrinsic flat sense to the round unit sphere $(\mathbb S^3,g_{rd})$. The conjecture asks whether nearly sharp scalar-curvature and min-max-width bounds, together with uniform diameter and volume bounds, force convergence to the round metric; the paper proves this conjecture under rotational symmetry and extends the result to dimensions $n\geqslant 3$.