Stabilized Lagrangian isotopy conjecture for smoothly ribbon-isotopic cobordisms

Let $L_1$ and $L_2$ be decomposable Lagrangian cobordisms that are smoothly ribbon isotopic. A stabilization of a Lagrangian cobordism is the stabilization described in the paper, obtained along a properly embedded arc joining its negative and positive Legendrian ends.

Stabilized Lagrangian isotopy conjecture. If $L_1$ and $L_2$ are smoothly ribbon isotopic, then after stabilizing both cobordisms, they become Lagrangian isotopic.

The conjecture proposes that stabilization removes the remaining symplectic rigidity between smoothly ribbon-isotopic decomposable Lagrangian cobordisms, in analogy with the flexibility obtained by stabilization for Legendrian knots. The authors expect it to follow from techniques similar to those used in their proofs, but no resolution is supplied here.