Scharlemann's conjecture on reducible refills of a genus-two handlebody

Let $W$ be a genus-two handlebody embedded in a 3-manifold $M$, and let $[?]$ and $[?]$ be essential discs in $W$ whose refills produce exteriors $M[\alpha]$ and $M[\beta]$. Suppose $(M,W)$ is admissible, meaning that every sphere in $M$ separates, $M$ has no lens-space connected summands, any two curves in $\partial M$ compressible in $M$ are isotopic in $\partial M$, $M-W$ is irreducible, and $\partial M$ is incompressible in $N$. Scharlemann's conjecture. If $(M,W)$ is admissible, then at least one of the following occurs: $M=S^3$ and $W$ is unknotted, at least one of $M[\alpha]$ and $M[\beta]$ is irreducible and boundary-irreducible, or $\alpha$ and $\beta$ are aligned in $W$. The conjecture describes the exceptional configurations in which both refilled exteriors can fail to be irreducible and boundary-irreducible; the source does not provide enough information here to determine its resolution status.