Finiteness conjecture for non-destabilizable Legendrian link types

Let $\mathscr L$ be a link type in $\mathbb S^3$. A Legendrian link type is non-destabilizable if it cannot be obtained from another Legendrian link type by a destabilization. Finiteness conjecture. For any link type $\mathscr L$ in $\mathbb S^3$, there are only finitely many non-destabilizable Legendrian link types having topological type $\mathscr L$. This is a general folklore conjecture motivated by the classification problem for Legendrian links; it is unsettled in general, although it is known for several specific knot types.