Two-minima one-saddle conjecture for alternating links

Let $L$ be an alternating link. A ribbon surface with two minima and a single saddle point is a ribbon surface for $L$ having exactly two minima and one saddle point. A link is algorithmically ribbon if it admits the algorithmic ribbon-disk construction defined earlier in the paper.

Two-minima one-saddle conjecture. If $L$ bounds a ribbon surface with two minima and a single saddle point, then $L$ is algorithmically ribbon.

The conjecture is motivated by experimental evidence for the algorithm and by a comparable theorem of McCoy concerning unknotting numbers of alternating knots. Its general validity remains open in the supplied source.