The disc braid-group conjecture for topological complexity

Let $D$ be the disc, let $C(D,n)$ be its unordered configuration space, and let $B_n$ be the braid group on $n$ strands. The disc braid-group conjecture asserts that

$$\TC(C(D,n))=\TC(B_n)=2n-3.$$

This would follow if the conjectured equality $\operatorname{cd}([P_n,P_n])=n-2$ gave the lower bound $\TC(C(D,n))\geq 2n-3$ and the stated upper bound extended to all $n$. The paper leaves this as a conjectural consequence and does not establish it for general $n$.