The weak stable Generalized Property R conjecture

Let $L$ be an R-link, and let $U$ be a 0-framed unlink. A Hopf pair is a Hopf link with one component 0-framed and the other dotted, representing a 4-dimensional 1-handle. The weak stable Generalized Property R conjecture. For every R-link $L$, there is a 0-framed unlink $U$ and a split collection of Hopf pairs $V$ such that $L\sqcup U\sqcup V$ is handleslide-equivalent to an unlink and a split collection of Hopf pairs. This is a further weakening of the Generalized Property R conjecture, motivated by stabilizing Kirby diagrams; its general validity is open.