Morse-Novikov genus conjecture for oriented links

For an oriented link $L$ in $S^3$, let $g_{{\rm{MN}}}(L)$ denote the minimum genus among possibly disconnected Seifert surfaces for $L$ that realize the Morse-Novikov number, and let $g(L)$ be the genus of $L$.

Morse-Novikov genus conjecture. One expects

$$g_{{\rm{MN}}}(L)=g(L).$$

This asserts that the Morse-Novikov genus is realized by a minimal-genus Seifert surface. The statement is not known in general, although the paper notes that for knots, and also for oriented links allowing disconnected Seifert surfaces, the Morse-Novikov genus is realized by an incompressible Seifert surface.