The stable Generalized Property R conjecture

Let $L$ be an R-link, meaning an $n$-component link in $S^3$ whose 0-surgery yields $Y_n=\mathop{\#}^n(S^1\times S^2)$. Let $U$ be a 0-framed unlink. The stable Generalized Property R conjecture. For every R-link $L$, there is a 0-framed unlink $U$ such that $L\sqcup U$ is handleslide-equivalent to an unlink. This is a weaker stabilization of the Generalized Property R conjecture and has the same implication for the associated homotopy 4-spheres; its general validity is open.