Set of points with a unique closest point in a compact set

Let $Ksubsetmathbb{R}^n$ be any compact set. Let $operatorname{Unp}(K)$ be the set of points in
$$
operatorname{Unp}(K)={xinmathbb{R}^nsetminus K:, exists ! yin K |x-y|=d(x,K)}.
$$
Here are some properties.

1. The distance function is Lipschitz and hence differnetiable a.e. (Rademacher’s theorem). If the distance function is differentiable at $x$, then $xin operatorname{Unp}(K)$. For a proof, see https://mathoverflow.net/a/299066/121665.
   Therefore almost all points of $mathbb{R}^nsetminus K$ belong to $operatorname{Unp}(K)$.

2. For every $xin mathbb{R}^nsetminus K$, there is $yin K$ such that $|x-y|=d(x,K)$ (although $y$ is not unique). Then the interior of the segment $xy$ is contained in $operatorname{Unp}(K)$ (triangle inequality).

Thus $operatorname{Unp}(K)$ contains the union of disjoint open segments and the set $mathbb{R}^nsetminus (operatorname{Unp}(K)cup K)$ is of measure zero and is contained in the endpoints of these segments.

Question. Is it true that $operatorname{Unp}(K)$ contains an open set?

The interest in the study of the properties of the distance function on the set $operatorname{Unp}(K)$ is motivated by results of Federer about sets of positive reach.

H. Federer,
Curvature measures.
Trans. Amer. Math. Soc. 93 (1959), 418–491.