White's sharp Hausdorff-dimension conjecture for cone stratifications

Let $\mathcal{C}$ be a compact family of closed cones. Suppose that at each point of a set $S$, each blow-up of $S$ is contained in some element of $\mathcal{C}$. Let $d$ be the largest possible Hausdorff dimension of $S$. White's sharp stratification conjecture. The number $d$ equals the largest dimension of a linear subspace that is a subset of one of the cones in $\mathcal{C}$. The conjecture would sharpen White's stratification theorem, which gives only that $d$ is at most the largest building dimension of the cones in $\mathcal{C}$. The source explicitly states that a counterexample, together with the stratification theorem, gives an indirect proof that this conjecture is not valid.