Conjecture on convex level sets in Alexandrov spaces

Let $X$ be an Alexandrov space with curvature bounded below by $\kappa$, and let $f:X\to\mathbb{R}$ be a convex function. For a real number $c$, consider the level set

$$\{f=c\}.$$

Conjecture on convex level sets in Alexandrov spaces. The level set $\{f=c\}$ is an Alexandrov space of curvature at least $\kappa$.

The source states this as an unknown claim widely expected to hold when the ambient Riemannian manifold in the preceding example is replaced by an Alexandrov space. The precise hypotheses and validity of the proposed generalization remain to be established.