The universal $(2,2)$-move reduction conjecture for links

A $(2,2)$-move is the local move shown in the source; a $5$-move is described there as a combination of $(2,2)$-moves and their mirror-image $(-2,-2)$-moves. Universal $(2,2)$-move reduction conjecture. Every link is $(2,2)$-move equivalent to a trivial link. This is presented as another elementary-moves question, but the supplied passage gives no proof or counterexample and does not state its resolution.