Knotted handlebodies in the 4-sphere and 5-ball

For every integer $g\ge 2$ we construct 3-dimensional genus-$g$ 1-handlebodies smoothly embedded in $S^4$ with the same boundary, and which are defined by the same cut systems of their boundary, yet which are not isotopic rel. boundary via any locally flat isotopy even when their interiors are pushed into $B^5$. This proves a conjecture of Budney-Gabai for genus at least 2.