Knotted handlebodies in the 4-sphere and 5-ball

For every integer $g\ge 2$ we construct smooth 3-dimensional 1-handlebodies $H_1$ and $H_2$ of genus $g$, which are properly embedded in $B^5$ with the same boundary, such that both $H_1$ and $H_2$ are smoothly boundary parallel and are homeomorphic rel boundary as 3-manifolds, yet $H_1$ and $H_2$ are not related by a topological, locally flat isotopy rel boundary. In other words, we construct genus-$g$ 3-dimensional 1-handlebodies smoothly embedded in $S^4$ with the same boundary that are defined by the same cut systems of their boundary yet are not isotopic rel boundary via any locally flat isotopy even when their interiors are pushed into $B^5$. This in particular proves a conjecture of Budney--Gabai for $g\ge 2$.