  Let $M$ be a simple 3-manifold, and $F$ be a component of $\partial M$ of
genus at least 2. Let $\alpha$ and $\beta$ be separating slopes on $F$. Let
$M(\alpha)$ (resp. $M(\beta)$) be the manifold obtained by adding a 2-handle
along $\alpha$ (resp. $\beta$). If $M(\alpha)$ and $M(\beta)$ are
$\partial$-reducible, then the minimal geometric intersection number of
$\alpha$ and $\beta$ is at most 8.
