The $2d-1$ threshold conjecture for collapsing unit vectors

Let $\overline{\mathcal{C}}(k,d)$ denote the extremal quantity for $k$-collapsing families of unit vectors in $d$-dimensional normed spaces. The $2d-1$ threshold conjecture.

$$\overline{\mathcal{C}}(k,d)=k+1\quad\text{whenever }k\geq 2d-1.$$

The preceding theorem proves the equality when $k$ is sufficiently large compared with $d$, while the text notes that the stated threshold is necessary because examples in $\ell_\infty^d$ give larger families for $k\leq 2d-2$. The conjecture is known in the supplied text for $d\leq 5$ and remains open in general.