The aligned meridians conjecture for admissible pairs

Let $(M,W)$ be an admissible pair, with $W$ a genus two handlebody and $\alpha,\beta$ the specified meridian disks of $W$; the source defines admissibility and alignment in the preceding text. Aligned meridians conjecture. If $(M,W)$ is admissible, then either

$$W\text{ is unknotted and }M=S^3,$$

or at least one of $M[\alpha]$ and $M[\beta]$ is both irreducible and $\partial$-irreducible, or $\alpha$ and $\beta$ are aligned in $M$. This is presented as the main conjecture of the paper, and the source gives no resolution.