The slice Akbulut–Kirby conjecture

Let $K$ and $K'$ be knots in $S^3$, and let $S^3_0(K)$ denote the $3$-manifold obtained by zero surgery on $K$. A knot is slice if it bounds a smoothly embedded disk in $B^4$. Slice Akbulut–Kirby conjecture. If $K$ is slice and there is a zero-surgery homeomorphism

$$\phi:S^3_0(K)\longrightarrow S^3_0(K'),$$

then $K'$ is also slice. The conjecture is the slice case left open by Yasui's counterexamples to the original Akbulut–Kirby conjecture, and it motivated the Manolescu–Piccirillo approach to the smooth Poincaré conjecture in dimension four.