Budney–Gabai conjecture on non-isotopic compressing curve equivalent handlebodies

Let $H_1$ and $H_2$ be 3-dimensional genus-$g$ handlebodies embedded in $S^4$, for an integer $g\geq 0$, with common boundary

$$\partial H_1=\partial H_2.$$

They are compressing curve equivalent if there exist $g$ disjoint simple closed curves $A_1,\ldots,A_g$ in their common boundary such that the complement of tubular neighborhoods is planar and each $A_i$ bounds disks in both $H_1$ and $H_2$. Budney–Gabai conjecture. For each $g\geq 0$, there exist such handlebodies $H_1$ and $H_2$ that are compressing curve equivalent, but $H_1$ is not isotopic to $H_2$ via an isotopy fixing their common boundary. The conjecture concerns the distinction between boundary data and isotopy classes of embedded handlebodies. The paper's abstract states that the conjecture is proved for every genus at least $2$, while the genus-$0$ and genus-$1$ cases are not resolved by the supplied text.