Neuwirth's conjecture for non-split links

Let $L$ be a non-trivial, non-split link, and let $E(L)$ denote its exterior. A closed surface $F$ containing $L$ has each component of $L$ non-separating in $F$ when no component separates $F$, and $F$ is essential in $E(L)$ when it is incompressible and boundary incompressible there. Neuwirth conjecture for links. For any non-trivial, non-split link $L$, there exists a closed surface $F$ containing $L$ such that each component of $L$ is non-separating in $F$ and $F$ is essential in $E(L)$. This extends the Neuwirth conjecture from knots to links; the supplied text gives no resolution status.