The band-sum unknotting-number conjecture

A band sum is the knot obtained by joining two knots by a band; the unknotting number $u(K)$ is the minimal number of crossing changes required to transform a knot $K$ into the unknot. A knot is nontrivial if it is not isotopic to the unknot.

Band-sum unknotting-number conjecture. Any band sum of two nontrivial knots has unknotting number greater than one.

Scharlemann's theorem proves the analogous statement for composite knots, but the band-sum formulation is a proposed strengthening and remains unresolved in the supplied source.