Suzuki's unknotting conjecture for 2-knots

A 2-knot is a smoothly embedded $2$-sphere in $S^4$; its equator is the intersection with an equatorial $S^3$, and an equator is unknotted when it is the standard unknot.

Suzuki's unknotting conjecture. Let $G$ be a $2$-knot in $S^4$ with unknotted equator. Then $G$ is smoothly unknotted.

The conjecture concerns whether a $2$-knot with an unknotted equatorial cross-section must itself be standard. The source notes that an affirmative answer to the preceding handlebody question would imply this conjecture.