The partial-conjugation relation conjecture for string links

For an $n$-component string link, let $\overline{x}_{ij}$ denote the partial conjugation associated with the ordered pair of distinct components $i,j$. Saying that an equality holds up to commutators means that it holds after disregarding commutator terms.

Partial-conjugation relation conjecture. For each $1\leq i\leq n$,

$$\prod_{j;\,j\neq i} \overline{x}_{ij}={\rm id}$$

holds up to commutators. Consequently, the number of generators of partial conjugations is reduced to $n(n-2)$.

The relation is verified in the 4- and 5-component cases, and similar relations can be checked for 2 and 3 components. The conjecture asserts the corresponding relation for arbitrary numbers of components.