Conjecture on reducing links using (2,2)-moves and sixth-power braid moves

A link is a disjoint union of circles in $S^3$. A $7(2,2)$-move is a $(2,2)$-move or its mirror image, and a $(\sigma_1\sigma_2)^6$ move is the local move associated with the sixth power of the standard braid word on three strands. Reduction conjecture. Is every link reducible to a trivial link by $7(2,2)$-moves and $(\sigma_1\sigma_2)^6$ moves? This is posed as a research problem motivated by B. Wajnryb; the supplied text gives no resolution.