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

I think the following is a counterexample in $\mathbb{R}^2$. Consider the curve whose polar coordinates expression is $r = \sum_{n=1}^\infty \frac{1}{a_n} \sin(2\pi a_n\theta)$, where $a_n = 100^n$, say. Let $K$ be this curve together with all points interior to it.

It seems to me that any point $x$ with a unique closest point $y$ in $K$ has arbitrarily close neighbors whose closest point is nonunique. Because in a small neighborhood of $y$ the curve looks like a little sine wave, and there will be a point $x'$ close to $x$ which is equidistant from two peaks near $y$. As you zoom in more you might need to adjust $x'$ slightly but it will still be close to $x$. Needs some work, but ...

There are many counterexamples as the following result of Zamfirescu [1] shows.

Theorem. For most of the compact sets $K\subset\mathbb{R}^n$, $\operatorname{Unp}(K)$ has empty interior, meaning that the set of points in $\mathbb{R}^n$ without a unique nearest point in $E$ is dense in $\mathbb{R}^n$.

Here "most of the compact sets" is understood in the Baire category sense with respect to the Hausdorff metric on the space of compact sets in $\mathbb{R}^n$.

[1] T. Zamfirescu, The nearest point mapping is single valued nearly everywhere. Arch. Math. (Basel) 54 (1990), 563–566.