Converse to the H(2)-unknotting characterization for composite links

Let $K_1$ be the 2-bridge link $S(q,p)$, and let $K_2$ be a $(p,q)$-tangle unknotting number one link. The composite link $K_1\sharp K_2$ then has H(2)-unknotting number one.

Converse conjecture. Every composite link with H(2)-unknotting number one has the form described in Proposition~.

This conjecture asks whether the sufficient construction for composite links with H(2)-unknotting number one is also necessary. The paper gives the construction but does not establish the converse.