Stable Property R conjecture for links

Let $U_n$ be the $n$-component zero-framed unlink in $S^3$. A link $L$ has Stable Property R if, for some integers $r$ and $m$, the split union $L\sqcup U_r$ is handleslide equivalent to $U_m$. Stable Property R conjecture. Every R-link has Stable Property R. This is the stable version of the Property R problem for links arising from surgery to a connected sum of copies of $S^1\times S^2$; the supplied text gives no resolution status.