Boninger–Greene conjecture on positive knots and ribbon concordance

Let $K_0,K_1\subset S^3$ be knots, and write $K_0\leq K_1$ when there is a ribbon concordance from $K_1$ to $K_0$. A knot is positive if it admits a diagram in which all crossings are positive. Boninger–Greene conjecture. If $K_1$ is a positive knot and $K_0\leq K_1$, then

$$K_0\cong K_1.$$

This says that positive knots are minimal in the ribbon-concordance order. The source proves the conjecture for a large class of positive knots, extending earlier results for special alternating knots, but the general statement remains open.