The sharp $σ_k$-curvature inequality on locally conformally flat manifolds in quantitative form

Let $2\leq k<n/2$ and let $(M^n,[g])$ be a smooth, closed, connected, and locally conformally flat Riemannian manifold with a $k$-admissible metric in the conformal class $[g]$. We prove a stability result of the $σ_k$-curvature inequality on $M$, in the sense that if equality is almost satisfied fo