The cohomological-dimension conjecture for commutator subgroups of pure braid groups

Let $P_n$ be the pure braid group of the disc, and let $[P_n,P_n]$ denote its commutator subgroup. The cohomological-dimension conjecture asserts that

$$\operatorname{cd}([P_n,P_n])=n-2.$$

This value would make the lower bound in the disc case of the topological-complexity theorem equal to $2n-3$, substantially narrowing the possible values of $\TC(C(D,n))$. The paper presents this as an expectation rather than a proved result.