Bernhard–Jablan unknotting conjecture

Let $K$ be a knot. A minimal-crossing-number diagram of $K$ is a diagram of $K$ with the minimum possible number of crossings. For such a diagram $D$, let ${\sf BJSet}_K$ be the set of knots obtained by changing a single crossing in a minimal-crossing-number projection of $K$, and define the strong Bernhard–Jablan unknotting number by

$$u_{BJ}^s(K)=1+\min\{u(K')\mid K'\in {\sf BJSet}_K\}.$$

Bernhard–Jablan conjecture. Every knot $K$ possesses a minimal-crossing-number diagram $D$ and a crossing in $D$ such that changing the crossing results in a diagram $D'$ for a knot $K'$ with $u(K')<u(K)$. Equivalently,

$$u(K)=\nu_{BJ}^s(K)$$

for all knots $K$.

The conjecture was proposed by Bernhard and Jablan as an attempt to compute unknotting number using minimal-crossing-number diagrams. It is false in general, so the claimed equality does not hold for all knots.