The $2d$ extremal-value conjecture for large collapsing parameters

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

$$\overline{\mathcal{C}}(k,d)=2d\quad\text{if }2d-\sqrt{d/2}\leq k\leq 2d-2.$$

The conjecture concerns the range not covered by the preceding $k+1$ threshold conjecture and has non-empty content only for $d\geq 8$, according to the source. The supplied text gives no resolution beyond the stated known cases.