I have two questions about the slice=ribbon conjecture.
(1) If a knot $K \hookrightarrow S^3$ has smooth slice genus $g$, you can ask if it bounds a smooth genus $g$ surface in $S^3 \times [0, -\infty)$, with the function defined by restriction to $[0, -\infty)$ being Morse on the surface without index=0 critical points (maximal points). When $g=0$ this is just asking if the slice knot $K$ has a ribbon disc. I was wondering if there are any knots known with $g \geq 1$ for which such a surface cannot exist. If there are none such known, is there a topological reason why the truth of the slice=ribbon conjecture would also imply the existence of such surfaces?
(2) Are there any potential counterexamples to slice=ribbon (in the same way that there are potential counterexamples to smooth 4-d Poincare [until Akbulut kills them])?