The extremal Borsuk number conjecture for normed spaces

Let $B(n)$ be the maximum, over all norms on $\mathbb{R}^n$, of the Borsuk number of $(\mathbb{R}^n,\left\Vert\cdot\right\Vert)$, where the Borsuk number is the smallest number of sets of strictly smaller diameter needed to partition every bounded subset. Normed-space Borsuk conjecture. For each integer $n\geq 3$,

$$B(n)=2^n.$$

This conjecture is motivated by the inequality $b_X(K)\leq c(K)$ and Hadwiger's covering conjecture, which predicts the bound $c(K)\leq 2^n$ for convex bodies. Despite progress, it remains open when $n\geq 3$.