White's building-dimension characterization for closed cones

Let $C$ be a closed cone in $\mathbb{R}^n$. Its building dimension is

$$\operatorname{bdim}(C):=\sup\{\dim_{\mathcal{H}}(S):S\subset\mathbb{R}^n,\ {\rm diff}(S)\subset C\},$$

where $\operatorname{diff}(S):=\{x-y:x,y\in S\}$. White's building-dimension conjecture. The building dimension of $C$ equals the supremum of the dimensions of the vector subspaces contained in $C$. This would extend the known characterization from convex cones and cones contained in $\mathbb{R}^2$ to all closed cones. The conjecture is refuted by a counterexample in the paper, which, together with White's stratification theorem, also yields an indirect disproof of the second conjecture considered there.