  Given $ n \geq 2 $ and $ k \in \{2, \ldots , 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^\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.
