Embedded annulus classification conjecture for minimal Lagrangian surfaces

Let an annulus-type surface be an embedded surface whose underlying domain is an annulus. Assume it is minimal and Lagrangian in the unit ball, with Legendrian capillary boundary on the unit sphere. Embedded annulus classification conjecture. Any embedded annulus-type minimal Lagrangian surface with Legendrian capillary boundary on $\mathbb{S}^3$ is one of the examples given in Example 2.7. This refines the preceding nonexistence conjecture by allowing capillary rather than free boundary and proposing a classification of all embedded annular examples; the source does not establish the classification.