Stability for barriers of n-dimensional convex bodies with surface area close to Jones' bound

Let $K$ be a convex body (a non-empty compact convex set) in $n$-dimensional Euclidean space. A set $B$ is called a barrier (or an `opaque set') for $K$ if every line that intersects $K$, also intersects $B$. Although this concept was introduced more than a century ago, the barrier with minimal surf