The boundary-irreducible aligned meridians conjecture

Let $(M,W)$ be an admissible pair, where $W$ is a genus two handlebody with meridian disks $\alpha$ and $\beta$, and suppose that $M-W$ is $\partial$-irreducible. Boundary-irreducible aligned meridians conjecture. Either at least one of $M[\alpha]$ and $M[\beta]$ is both irreducible and $\partial$-irreducible, or $\alpha$ and $\beta$ are aligned in $M$. By the proposition cited in the source, this formulation is equivalent to the preceding main conjecture; the source gives no resolution.