The finite-predecessor conjecture for ribbon concordance

Let $K$ be a knot in $S^3$, and write $J\leq K$ when the knot $J$ is ribbon concordant to $K$. Finite-predecessor conjecture. For each knot $K\subset S^3$, there are only finitely many knots $J\leq K$.

This conjecture was proposed as an a priori stronger version of Gordon's descending-chain conjecture. The paper's abstract states that the authors prove the corresponding finiteness result for fibered predecessors, while the unrestricted conjecture is not stated as resolved here.