The genus-g handlebody detection conjecture in the 4-sphere

For each $g\geq 0$, let $V_0,V_1\subset S^4$ be 3-dimensional genus-$g$ handlebodies with the same boundary, $\partial V_0=\partial V_1$. A simple closed curve in $\partial V_0$ is $V_0$-compressible if it bounds a compressing disc in $V_0$. Handlebody detection conjecture. For every $g\geq 0$, there exist $V_0$ and $V_1$ such that the sets of $V_0$-compressible and $V_1$-compressible simple closed curves coincide, but $V_1$ is not isotopic to $V_0$ through an isotopy fixing $\partial V_1$. This asks whether the compressible-curve data on the common boundary completely detects the embedding of a handlebody in $S^4$.