$partial$-reducible handle additions

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.