The sliceness conjecture for 2-bridge knots

For a knot $K\subset S^3$, let $g_{4}(K)$ and $g_{4}^{{\rm top}}(K)$ denote its smooth and topological slice genera, respectively. A knot is smoothly slice when $g_{4}(K)=0$ and topologically slice when $g_{4}^{{\rm top}}(K)=0$.

Sliceness conjecture for 2-bridge knots. For all $2$-bridge knots $K$, $g_{4}(K)=0$ if and only if $g_{4}^{{\rm top}}(K)=0$.

This conjecture asks whether smooth and topological sliceness coincide for 2-bridge knots. The paper disproves its natural generalization from sliceness to higher slice genera, while the stated sliceness question for 2-bridge knots is the motivating distinction explored in the paper.