Kohn's unknotting-crossing conjecture for minimal diagrams

Let $K$ be a knot, and let $u(K)$ denote its unknotting number. An unknotting crossing is a crossing whose change produces the unknot, and a minimal diagram is a diagram of $K$ with the minimum possible number of crossings.

Kohn's conjecture. If

$$u(K)=1,$$

then $K$ has an unknotting crossing in a minimal diagram.

This conjecture has been resolved in a number of cases, as stated in the source; the supplied status evidence therefore records it as solved.