Bergström's conjecture on the Euclidean Steinitz constant

Let $B_2^d$ denote the Euclidean unit ball in $\mathbb{R}^d$, and let $S(B_2^d)$ be its Steinitz constant: the smallest $C$ such that every finite set of vectors $V\subset B_2^d$ with sum zero admits an ordering whose partial sums have Euclidean norm at most $C$. Bergström's conjecture. For all $d\geq 1$,

$$S(B_2^d)=O(\sqrt{d}).$$

This conjecture, known as the Steinitz problem, sought a much stronger estimate than the general bound $S(B)\leq d$. The supplied status evidence says that it has been refuted.