I will use the terminology of this paper by A. Petrunin: https://arxiv.org/pdf/1304.0292.pdf

Let a sequence of $n$-dimensional Alexandrov spaces $\{X_i\}$ of curvature at least $-1$ converges to an Alexandrov space $X$ in the Gromov-Hausdorff sense. Let $f_i\colon X_i\to \mathbb{R}$ be 1-Lipschitz $\lambda$-concave functions (see Def 1.1.1 and 1.1.2 on p.5 of the above paper) converging to a function $f\colon X\to \mathbb{R}$.

**Is the function $f$ also $\lambda$-concave?**

I think the question has the positive answer if $X$ has no boundary. My question is about the case when $X$ has a non-empty boundary.