Finiteness conjecture for non-simplifiable rectangular diagrams

Let $\mathscr L$ be a link type in $\mathbb S^3$. A rectangular diagram representing $\mathscr L$ is non-simplifiable if it cannot be reduced by the simplifying moves considered for rectangular diagrams. Finiteness conjecture. For any link type $\mathscr L$, there are only finitely many non-simplifiable rectangular diagrams representing $\mathscr L$. The paper states that this conjecture is equivalent to the finiteness conjecture for non-destabilizable Legendrian link types, via the commutation property for type-I and type-II moves; no resolution is supplied here.