Finite total curvature and soap bubbles with almost constant higher-order mean curvature

Given $n\ge 2$ and $k\in \{2,\dots ,n\}$, we study the asymptotic behaviour of sequences of bounded $C^2$-domains of finite total curvature in ${\mathbb{R}}^{n+1}$ converging in volume and perimeter, and with the $k$-th mean curvature functions converging in $L^1$ to a constant. Under natural mean convexity hypothesis, and assuming an $L^{\mathrm{\infty }}$-control on the mean curvature outside a set of vanishing area, we prove that finite unions of mutually tangent balls are the only possible limits. This is the first result where such a uniqueness is proved without assuming uniform bounds on the exterior or interior touching balls.