Fixed-level bridge finiteness conjecture for links in the 3-sphere

For an integer $n\ge 1$, an $n$-bridge position of a link in $S^3$ is a bridge presentation with $n$ bridges, considered up to bridge isotopy. Fixed-level bridge finiteness conjecture. For each integer $n\ge 1$, every link type in $S^3$ admits at most finitely many bridge isotopy classes of $n$-bridge positions. This conjecture asks for finiteness at each fixed bridge number and is a finer question than finiteness of link types with bounded bridge number; its status is not determined by the supplied source.