Finiteness conjecture for Lagrangian ribbon disk fillings

Let $L$ be a Legendrian ribbon knot in $(S^3,\xi_{\text{st}})$. A Lagrangian ribbon disk is a Lagrangian disk in $B^4$ with ribbon singularities. Finiteness conjecture. Then $L$ bounds only finitely many Lagrangian isotopy types of Lagrangian ribbon disks in $B^4$. The conjecture proposes a finiteness phenomenon for ribbon disk fillings, in contrast with the existence of Legendrian knots admitting arbitrarily many distinct smooth isotopy classes of Lagrangian disk fillings; its resolution is not established in the supplied text.